# Marching with the Pink Parade: Evaluating Visual
Search Recommendations for Non-binary Clothing Items

Siddharth D Jaiswal<EMAIL_ADDRESS>and  Animesh Mukherjee
<EMAIL_ADDRESS>Dept. of Computer Science and Engineering, Indian
Institute of TechnologyKharagpurWest BengalIndia

###### Abstract.

Fashion, a highly subjective topic is interpreted differently by all
individuals. E-commerce platforms, despite these diverse requirements, tend to
cater to the average buyer instead of focusing on edge cases like non-binary
shoppers. This case study111See video demo here: https://tinyurl.com/ddv886h2,
through participant surveys, shows that visual search on e-commerce platforms
like Amazon, Beagle.Vision and Lykdat, is particularly poor for non-binary
clothing items. Our comprehensive quantitative analysis shows that these
platforms are more robust to binary clothing inputs. The non-binary clothing
items are recommended in a haphazard manner, as observed through negative
correlation coefficients of the ranking order. The participants also rate the
non-binary recommendations lower than the binary ones. Another intriguing
observation is that male raters are more inclined to make binary judgements
compared to female raters. Thus it is clear that these systems are not
inclusive to the minority, disadvantaged communities of society, like LGBTQ+
people. We conclude with a call to action for the e-commerce platforms to take
cognizance of our results and be more inclusive.222This work has been accepted
for publication at ACM CHI 2022 (Case Studies) as an Extended Abstract

e-commerce, visual search, gender neutrality, recommendation systems

††copyright: none

##  1\. Introduction

Fashion permeates every aspect of modern human life, ranging from clothing and
lifestyle to culture and societal behaviour. People shop for clothes, apparel,
and footwear from both brick and mortar stores and online e-commerce
platforms. The fashion industry, projected to be worth $3.3 trillion by 2030
(Remotely, 2021), is a substantial part of the global economy and affects many
livelihoods in the production and supply chains.

Although there has been a lot of stigma against LGBTQ+ people historically and
almost no concern for their rights in any aspect of societal life, in recent
time there is a growing understanding and acceptance of gender fluidity and
homosexuality (Samanwaya Rautray, 2018).

As with other commodities and services (for e.g., access to public washrooms,
changing rooms in clothing stores, etc), clothing items have been segregated
on the basis of binary genders – male and female with no recognition for trans
or other gender fluid needs or preferences. LGBTQ+ people require safe spaces
in shared public areas to deter harassment and discrimination (Weinhardt et
al., 2017), which such establishments may not be able to provide. Thus, a
person from a disadvantaged community might prefer shopping for clothes and
other items online from the safety and comfort of their homes.

A huge gap exists in this space as none of the major e-commerce platforms
offer a non-binary option when shopping for clothing items or apparel (Amazon,
2021b; Flipkart, 2021), which means that an LGBTQ+ person has no option but to
search for clothes in the Men’s section or the Women’s section. Also, there is
no way for the said person to verify the fit or size of the clothing item as
the clothes are worn by cis-gendered models and aren’t representative of their
needs. Thus, we observe a clear need of bridging this gap by the e-commerce
platforms such that they can provide a holistic shopping experience to LGBTQ+
members of society by not just selling clothing items and apparel relevant to
them but also improve their shopping experience.

###  1.1. Visual search

Various online platforms and services support a special form of search called
visual search which allows a user to upload an image and get recommendations
relevant to the uploaded image. This can be used for searching for normal
items like, Google Lens (Google, 2021) and Pinterest Lens (Pinterest, 2021b)
or specifically for clothing items, like Amazon’s StyleSnap (Amazon, 2021a).
Thus visual search provides an extra dimension for searching items for users
who might not be able to provide the exact textual phrase or if text-based
search is not available in their preferred language. Similarly, visual search
may show more relevant results as the ML/DL model running in the backend has
access to more visual features it can use as compared to textual features.
Unfortunately, none of the models in use by commercial platforms are available
publicly and there is no definitive way of determining how the search is
working for any given service. Thus, there is a potential for various forms of
bias (Mehrabi et al., 2021) like evaluation bias, population bias, behavioural
bias, user interaction bias and emergent bias amongst others.

In this study, we explore possible user interaction and emergent biases in
visual search for various clothing items, including and specially gender-
neutral ones, by using expert and normal user opinions to identify whether the
search results are relevant to the input images on a Likert-like scale and
also what the rater’s perception is about the gender relevance of the
recommended clothing item (either male/female or non-binary). We perform this
study on three e-commerce platforms that allow visual search – Amazon
StyleSnap (Amazon, 2021a), Beagle.Vision (Beagle, 2021) and Lykdat (Lykdat,
2021).

###  1.2. Research questions

In this section, we state the research questions we will attempt to address in
this case study.

RQ1 Whether the raters’ consider the recommendations for the different types
of input clothing items to be binary or non-binary? Through this question, we
our intent is to identify how users’ perceive the gender of clothing and
whether the general opinion is inclined towards a non-binary or a binary
categorization.

RQ2 How relevant the recommendations are for any given input clothing item?
Through this question, we attempt to understand how well the visual search
recommendation system works for each of the three platforms identified above,
using a Likert-like scale. We use the recommendation scores to identify the
relevance across different types of gendered clothing items, and also to rank
the three platforms themselves in order of best to worst. We further check
whether the recommendation scores of the items are correlated to their rank in
the search results.

RQ3 How are the male raters’ responses in comparison to the female raters’
responses? Specifically, we study how the males respond to questions
pertaining to both gender and relevance for the given clothing items vs. how
the females respond to them and attempt to understand their opinions about the
results of the visual search on the three platforms.

##  2\. Literature Survey

We discuss some of the relevant prior literature studying fashion from a
social computing perspective as well some of the existing work in visual
search in recommendation systems.

###  2.1. Fashion in computing

Design and style being subjective topics, make it inherently difficult to
design systems that can define what is fashionable. Therefore, most research
involves human annotators. A recent work by Ragot et al. (Ragot et al., 2020)
used this to compare art generated by humans and AI. The social computing
community has also made attempts to understand what type of designs or fashion
appeal to humans. Park et al. (Park et al., 2016) evaluated the success of
models in the fashion industry based on their presence on Instagram.
Similarly, Kiapour et al. (Kiapour et al., 2014) were able to link personal
style with clothing behaviour. Yamaguchi et al. (Yamaguchi et al., 2014)
evaluated visual behaviour on fashion social networks and Maity et al. (Maity
et al., 2019) analyses the content in images on Pinterest fashion boards to
identify the driving factors for their popularity.  While none of these works
focus on non-binary clothing being sold in the market, our focus is on
precisely this because we believe that the existing system is far too broken.
This was seen in a recent related study by Dash et al. (Dash et al., 2021) who
identified a significant amount of exposure bias in the recommendations by
Amazon. Our study takes inspiration from this work and we feel that a careful
study should be performed on such systems and these should be fixed before new
technology can be installed. For example, Devendorf et al. (Devendorf et al.,
2016), in their CHI’16 work, explored how dynamic textile displays are
affected by personal style choices while a Pan et al. (Pan et al., 2012)
attempt to reconceptualize fashion in the domain of sustainable HCI. More
recently, works (Sun et al., 2021; Tepe, 2021) have appeared in CHI’20
exploring various methods for fashion design using novel methods.

In this case study, we identify that the industry and users as a whole need to
take a step back and evaluate the existing biases in genderfluid fashion
choices before designing for the future. Thus our work sits as a parallel to
the existing work on fashion in computing.

###  2.2. Visual Search

Visual search is found on search (Google, 2021), image-sharing (Pinterest,
2021b) and social media (Snapchat, 2021) applications. Lately, retailers have
been using it to enhance the shopping experience of customers on e-commerce
platforms (Shopify.in, 2021; Curatti.com, 2021; Synced, 2021). Amazon’s
Stylesnap (Amazon, 2021a) feature, launched in 2019, allows users to upload
images of clothing items and receive relevant recommendations for the same.
Beagle.Vision (Beagle, 2021) and Lykdat (Lykdat, 2021) work in a similar way
but none of these platforms have released their model architectures or
training datasets. This impedes a better understanding of their algorithms and
raises concerns of privacy and fairness (SmartInsights.com, 2021).

In this work, we attempt to investigate such fairness concerns. To this end,
we use images of individuals wearing different types of clothing items. These
individuals may belong to different gender groups.

##  3\. Dataset and Methodology

In this section we first discuss the details of the dataset used for the
study, followed by a description of the platforms evaluated as part of the
study. We next describe the demographics of the participants and finally the
format of collecting and evaluating user responses.

###  3.1. Dataset

Gender Category | Type of Clothing Item
---|---
Male-centric | bottomwear — shirt — T-shirt — suit
Female-centric | bottomwear — shirt — top — dress — formalwear
Non-binary | casualwear — dress — dungarees — suit — formalwear

Table 1. Type of clothing items collected per gender category in the dataset.
There are 8 unique items for each type leading to a total of 112 items.

Age | M | F | Professional Background
---|---|---|---
19-25 | 5 | 6 | Undergraduate College Student
26-30 | 1 | 2 | Research Scholar & Working Professional
31-35 | 1 | 0 | Working Professional

Table 2. Demographics of the participants who took part in the survey. The
majority were undergraduate college students and females.

(a)

(b)

(c)

Figure 1. Images collected from Pinterest website. (a) is an example of a
male-centric shirt, (b) is an example of a female-centric bottomwear, (c) is
an example of a non-binary suit.

We collected 112 images of various forms of clothing items from the Pinterest
(Pinterest, 2021a) website by searching for specific queries like ‘mens
jeans’, ‘womens tops’ and ‘androgynous clothing’ between July and September,
2021. We then distributed the collected clothing items into three categories
through a manual inspection of the data followed by a detailed discussion
between the authors and adjudication by a fashion design expert. The resolved
categories were male-centric clothing, female-centric clothing and non-binary
clothing. Clothes in each gender category were further distributed into
buckets corresponding to different clothing types – ‘bottomwear’, ‘T-shirts’,
‘suits’, etc. More details regarding the categories and count of images per
category are noted in Table 2 and images of some of the clothing items
collected from Pinterest are shown in Figure 1.

###  3.2. Platforms

(a)

(b)

(c)

(d)

Figure 2. Search results from the three platforms and a snapshot of the Google
form shared with the participants. (a) is for Amazon Stylesnap, (b) is for
Lykdat, (c) is for Beagle.Vision. Amazon recommends male centric clothing with
shirts and suits mixed together whereas both Beagle and Lykdat recommend both
male and female centric clothing. (d) is a snapshot of one of the Google forms
with the two questions for every output image.

Three platforms were evaluated for their visual search recommendations as part
of this case study – Amazon StyleSnap (Amazon, 2021a), because of its
pervasiveness in the consumer industry, and Beagle.Vision (Beagle, 2021) and
Lykdat (Lykdat, 2021) because, despite being relatively lesser known
companies, their visual search tech is openly accessible and is being used by
customers all over the world. Each platform allows a user to upload an image
of a clothing item and receive relevant recommendations for items that they
can either buy on the same platform (like Amazon) or are presented as out-
links to the corresponding seller’s website (Beagle.Vision and Lykdat). All
the three platforms detect multiple clothing items in the uploaded image and
show the recommendations for each such item, with both Beagle.Vision and
LykDat also displaying an associated label for each detected item. This
improves explainability as it allows a user to interpret the platform’s
understanding about the clothing item in question. On the other hand, since
Amazon does not share any labels, a user has no way of knowing the reason for
irrelevant recommendations. During our data collection, we discovered that
Amazon did not identify dungarees worn by males; instead the service retrieved
recommendations for overalls worn by shipyard workers and beekeepers. However,
the service perfectly identified the dungarees worn by females. We point out
here that these results are anecdotal and dependent on various factors like
the IP address, the time and day of the search and the browser on which the
search is performed and as such are subject to change with time and geographic
location. One sample result for each platform on a non-binary suit as input is
shown in Figure 2. As can be clearly seen, Amazon’s recommendations are
heavily male centric whereas both Beagle and Lykdat have female centric
recommendations as well.

###  3.3. Demographics of the survey participants

We advertised a call for volunteers on various social media sites like
LinkedIn and Twitter, and internal channels and were able to recruit 15 people
who consented to participate in the study. Eight of them self-identified
themselves as female and the other seven self-identified themselves as male.
The participants belong to various demographics ranging from undergraduate
college students to working professionals between the ages 19-35. One of the
participants was classified as an expert as the person had a background in
fashion design whereas the other participants were classified as normal users
who could be potential buyers of clothing items on an e-commerce platform. A
more detailed distribution of the participant demographics is noted in Table
2.

###  3.4. Data collection and participant surveys

We divided our study into two parts – data collection, performed between July
and September, 2021 and participant surveys, conducted in September, 2021.

Data collection: We automated our data collection procedure by using the
selenium tool to upload the input images one by one, and collect all the
recommendation responses for each input, on the Google Chrome browser from the
same geographic location and the same IP address. For each input image, we
collect the photo and the link to the product page displayed on the result
page, in the order it is displayed. We carried out this procedure for all the
three platforms. Note that these platforms are constantly changing their
designs as well as (possibly) their recommendations which might slightly
affect the results reported here.

Participant surveys: We created Google forms for our survey questions that
were then shared with the participants to fill. Each form took $\sim 2$ hours
to fill and had the following components.

  * •

Questions on personal details like name, email, age group, and consent. None
of these were made available to anyone except the authors.

  * •

28 unique input images randomly chosen from our collected dataset, ensuring an
equal number of images from each sub-category mentioned in Table 2.

  * •

For each input image, images of the top 5 products in the recommendation
result (henceforth referred to as output image), for each platform leading to
a total of 15 output images arranged in a random order. If any item was not
identified or there was no relevant recommendation due to misclassification,
we did not include any results for that particular platform, thus leading
lesser output images (5/10 images) for that input clothing item.

  * •

For each output image, there were two questions in the form – the first was an
MCQ asking the participant to choose according to their personal judgement
which gender the output image clothing most conformed to – male only, female
only or non-binary. The second question asked the participant to rate the
recommended image on its relevance using a 5-point Likert like scale with
rating points from 1 (least relevant) to 5 (most relevant). The exact
questions were as follows (Figure 2(d)).

    * –

Q1. Which of the following gender options is the most relevant for the given
piece of clothing?

    * –

Q2. Please choose the relevance score for your chosen recommendation.

##  4\. Results and Discussion

We aggregate the responses to the two questions per output image by each
participant and analyse the results for 112 input images and 1625 total output
images. We evaluate the participant responses across the dimensions of the
clothing categories created earlier – male-centric, female-centric and non-
binary across all platforms and also per online platform – Amazon Stylesnap,
Beagle.Vision and Lykdat.

###  4.1. Gender choice for clothing recommendations

| All | Stylesnap | Beagle.Vision | Lykdat
---|---|---|---|---
Male-centric |  Male only (49.92$\%$) |  Male only (56.22$\%$) |  Non-binary (48.04$\%$) |  Male only (50.90$\%$)
Female-centric |  Female only (63.03$\%$) |  Female only (68.42$\%$) |  Female only (56.17$\%$) |  Female only (64.14$\%$)
Non-binary |  Non-binary (47.42$\%$) |  Non-binary (49.15$\%$) |  Non-binary (45.34$\%$) |  Non-binary (47.81$\%$)
Table 3. Most frequent choice of gender option for recommended clothing items
aggregated over all input images and all participant responses. The rows
correspond to the gender option chosen by us for the input clothing items and
the individual cells correspond to the most frequent value chosen by the
participants as well as their percentages.

We first evaluate the participant answers for the gender they feel is most
relevant for the recommended image given an input image. The participant can
choose one of three options – male only, female only or non-binary. In Table
3, we present the results for all inputs distributed as per Table 2. Let us
denote the set of images in our dataset by $D$. The results are calculated in
the following manner – we first take the modal value from all participant
responses for each recommended image, thus getting a set of gender values
$G_{i}$ for each input image $i\in D$. Then we take the modal values from each
set $G_{i}$ and get the most frequent gender value response for each input
image $i\in D$. Finally, we consider all inputs distributed in the three
categories and calculate the most frequent gender choice per category. The
same procedure is followed for the modal values from each platform. The rows
in the table correspond to the gender option chosen by us for the input
clothing items. The individual cells correspond to the most frequent value
chosen by the participants along with their percentages.

Observations: From the second column (titled “All”), we can see that roughly
$50\%$ responses correspond to the gender category of the input image. This
confirms our own assumptions about the category divisions of the input
dataset. In the columns corresponding to each platform we see that the
percentage values are even more enhanced for each category of clothing with
more people being confident about the gender choice for female-centric
clothing as compared to male-centric or non-binary clothing. Interestingly,
for Beagle.Vision we see that a majority of the responses point toward the
clothing being non-binary instead of male-centric.

Takeaways: The following conclusions can be drawn from these results.

  * •

Amazon has the highest percentage of responses for the corresponding most
frequent gender options compared to Beagle.Vision and Lykdat.

  * •

The participants choose the non-binary option for the male-centric clothing
recommendations from Beagle.Vision most often. Thus this platform may appeal
more to people who are looking for non-binary clothing options.

  * •

The responses for Amazon Stylesnap and Lykdat are quite similar.

###  4.2. Average rating for the recommended output

| All | Stylesnap | Beagle.Vision | Lykdat
---|---|---|---|---
Male-centric |  3.89 — 0.34 |  3.65 — 0.83 | 3.54 — 1.06 |  4.09 — 0.41
Female-centric | 3.74 — 0.41 | 3.43 — 0.63 |  3.64 — 0.77 | 4.06 — 0.44
Non-binary | 3.35 — 0.50 | 2.75 — 1.08 | 3.33 — 0.96 | 3.39 — 1.19
Table 4. Mean relevance rating and the standard deviation for recommended
clothing items aggregated over all input images and all participant responses.
The rows correspond to the gender option chosen by us for the input clothing
items and the individual cells correspond to the mean ($\mu$) and standard
deviation ($\sigma$) per dataset category averaged over input images. The
highest value for each column are in bold.

We next evaluate the average ratings given by the participants to each
recommended item image on its relevance to the input image. The participants
rate each recommended image on a 5-point Likert like scale ranging from 1
(least relevant) to 5 (most relevant). The results for this experiment are
presented in Table 4 and calculated as follows- We first take the average
rating from all participant responses for each recommended image, thus getting
a set of average ratings for each recommended image. Next we average these
values across all the recommended images to obtain an average per input image
$i\in D$. Finally we calculate the mean ($\mu$) and standard deviation
($\sigma$) for each dataset category to obtain the final values.

Observations: From the table we observe that the male-centric clothing items
are the most relevant overall with the non-binary clothing items being the
least relevant. This confirms our assumption that online shopping platforms,
while being adept at recognizing and recommending relevant clothing items for
binary genders, are still lagging behind when it comes to recommendations for
non-binary clothing items. These differences are even more pronounced on the
Amazon platform, with non-binary clothing items receiving a recommendation of
only 2.75 as compared to 3.65 for male-centric clothing, a gap of almost 1
rating point. The ratings on Beagle.Vision are very close for the male and
female-centric clothing items with non-binary items being off by 0.31 rating
points from the female-centric items. Finally, we see that Lykdat has the
highest average ratings amongst all the three platforms for all input
categories, with male-centric clothing receiving an average rating as high as
4.09 out of 5. Although the average rating for non-binary clothing is highest
amongst all platforms at 3.39, it is still 0.70 points lower than the binary
gender items.

Takeaways: The following conclusions can be drawn from these results.

  * •

The average ratings for male-centric clothing are the highest and for non-
binary clothing are the least.

  * •

Although the non-binary gender choice is most frequent for Amazon (Table 3),
it has the lowest ratings for non-binary clothing amongst all categories and
platforms. This means that although it shows clothes that may be more non-
binary but they are not very relevant to the input being shown.

  * •

Similarly, Lykdat has the highest average ratings for all categories, with the
values for male-centric clothing being the highest amongst all categories and
platforms, even though it has a lower percentage of responses choosing the
male only option for its recommendations as compared to Amazon.

###  4.3. Ground truth ranking order vis-a-vis participant ranking order

| Stylesnap | Beagle.Vision | Lykdat
---|---|---|---
Male-centric | 0.102 | 0.080 | -0.038
Female-centric | 0.007 | 0.106 | -0.115
Non-binary | -0.003 | -0.040 | 0.114

Table 5. Average Spearman’s rank correlation between the ground truth rank
ordering and the participant rank ordering. The rows correspond to the gender
option chosen by us for the input clothing items and the individual cells
correspond to average Spearman’s rank for the corresponding platform. The
highest positive values for each column are in bold.

| Stylesnap | Beagle.Vision | Lykdat
---|---|---|---
Male-centric | 0.091 | 0.058 | -0.033
Female-centric | -0.004 | 0.088 | -0.095
Non-binary | -0.019 | -0.044 | 0.106

Table 6. Average Kendall’s $\tau$-b rank correlation between the ground truth
rank ordering and the participant rank ordering. The rows correspond to the
gender option chosen by us for the input clothing items and the individual
cells correspond to average Kendall’s $\tau$-b rank for the corresponding
platform. The highest positive values for each column are in bold.

| Stylesnap | Beagle.Vision | Lykdat
---|---|---|---
Male-centric | 0.602 | 0.676 | 0.550
Female-centric | 0.513 | 0.419 | 0.431
Non-binary | 0.292 | 0.259 | 0.618

Table 7. Average Spearman’s rank correlation between the expert’s rank
ordering and the normal participant rank ordering. The rows correspond to the
gender option chosen by us for the input clothing items and the individual
cells correspond to average Spearman’s rank for the corresponding platform.
The highest positive values are in bold.

| M-centric item | F-centric item
---|---|---
M participant |  M only (75.91$\%$) |  F only (86.37$\%$)
F participant |  M only (49.46$\%$) |  F only (68.74$\%$)

Table 8. Confusion matrix of participant’s gender and their response for the
gendered clothing items. The values in the cell describe the choice and the
percentage of times the particular gender choice is chosen by the participants
of that particular gender. Males have a more binary outlook in their choice of
gender for the clothing recommendations compared to females. M: male, F:
female.

We use the participant ratings to calculate the ranks for the recommended
outputs. The participants are shown the output images in a random order and
asked to rate each image on its relevance. This rating is used to calculate a
new ranking for each product, on a per platform basis. A higher average rating
means a lower (better) rank and vice versa. We then calculate the Spearman’s
correlation and Kendall’s $\tau$-b correlation by comparing the original
ground truth ranking against the participant ranking order.

We also find the correlation between the rank ordering by the expert
participant and a set of normal participants for a set of 28 input images and
their corresponding 405 output recommendations. This is done to understand how
the perspective of an expert differs from a normal user on a given visual
search recommendation result.

The results for the Spearman’s correlation are shown in Table 6, and the
results for Kendall’s $\tau$-b correlation are in Table 6. Finally, Table 8
presents the correlation coefficients for the comparison between the expert
and the normal user.

Observations: From Table 6 and Table 6 we can see that the correlations follow
a very distinct pattern. Amazon’s Stylesnap platform has the highest positive
correlation for male-centric clothing whereas it has negative correlations for
both female-centric and non-binary clothing items (Table 6). Similarly,
Beagle.Vision has the highest positive correlation for female-centric clothing
items and negative correlation for non-binary items, and a lower positive
correlation for the male-centric ones. Finally, Lykdat has the highest
positive correlation for non-binary clothing items, and negative correlations
for both male and female-centric clothing items. This is in stark contrast
with the results in Table 4 where Lykdat reported the highest average ratings
for all input clothing categories. More specifically, even though the average
ratings for male and female-centric clothing were $>4$, the correlations are
negative. This implies that even though the recommendations shown by Lykdat
are most relevant, their ranking order is very poor and completely opposite to
what the users would expect.

In Table 8, we compare the ranking done by the expert participant with the
rankings done by three other normal participants chosen randomly and calculate
the Spearman’s correlation coefficient for the same. Due to resource
limitations, we were unable to survey a higher number of participants. The
average correlations here are higher than that observed for the overall set of
images. Surprisingly, we see that Amazon has the highest positive correlation
for female-centric items whereas Beagle.Vision has the highest positive
correlation for male-centric items. Non-binary items have the highest positive
correlation on Lykdat, as before.

Takeaways: We can make the following conclusion from the results discussed
above.

  * •

Although Lykdat has the highest average ratings for all types of clothing
items, it also has least correlation for both male and female-centric clothing
items.

  * •

Beagle.Vision has the most negative correlation for non-binary clothing items,
despite having a higher average rating than Amazon Stylesnap for the same.

  * •

The comparison between expert and normal participant shows that the
correlation for Lykdat on non-binary items is very strong, but the results for
Amazon and Beagle.Vision are interchanged for male and female-centric items
respectively.

###  4.4. Female vs male participants

In this section, we compare the responses from the male and female
participants. The participants choose one of the three gender options for each
recommended item’s image. For the results noted in Table 8, we perform the
following steps to get the results - for each output image, we first calculate
the modal response from both male and female participants. We then count the
number of times each gender group selects ‘male only’ and ‘female only’ option
across all input images per clothing category. Finally, to calculate the
percentage values, we take a ratio of the count calculated previously with the
total number of output images within that category.

Observations: From the table we observe that male participants choose the
‘male only’ option $\sim 76\%$ of the times for the male-centric clothing
compared to female participants who do it only 40$\%$ of the times. Similarly,
male participants also choose the ‘female only’ option $\sim 86\%$ of the
times whereas female participants do it only 59$\%$ of the times.

On further investigation, we also observe that female participants choose the
‘non-binary’ option for male-centric clothing $\sim 76\%$ of the times,
against only $\sim 37\%$ of the times by male participants. Similarly, for
female-centric clothing, female participants choose the ‘non-binary’ option
$\sim 75\%$ of the times, as opposed to $\sim 26\%$ of the times by male
participants. These differences are very significant and show a stark contrast
in the behaviour of the male participants compared to the females.

Takeaways The following conclusions can be drawn from these results.

  * •

Males tend to choose the binary gendered option – ‘male only’ and/or ‘female
only’ more frequently than females.

  * •

Females choose the ‘non-binary’ option more frequently for both male-centric
and female-centric clothing as compared to the corresponding gender option,
thus indicating that females interpret clothing as non-binary more often.

  * •

This is an example of user interaction bias (Mehrabi et al., 2021) as the
observations can be attributed to the self-selected biased behaviour from the
users.

##  5\. Overall insights pointing to deprivation of the non-binary users

In this case study, we have evaluated three different e-commerce platforms –
Amazon, Beagle.Vision and Lykdat for their visual search recommendations and
evaluated these recommendations through human surveys. With growing public
debate surrounding LGBTQ+ rights, there has been a rising concern about
creating an inclusive shopping experience for LGBTQ+ shoppers (YouthKiAwaaz,
2021) and various fashion retailers have adopted or pivoted themselves to
address this concern (FIBRE2FASHION, 2021; CNBC, 2021). While this has worked
for a few e-commerce platforms (Vue.ai, 2021), large online marketplaces like
Amazon have refrained from adopting strategies as proposed by (Vue.ai, 2021;
CONTENTSQUARE, 2021).

We have presented a comprehensive quantitative evaluation of the human surveys
over recommendations for male-centric, female-centric and non-binary clothing
items and made some important observations. Based on the results in Tables 3
and 4, it is immediately clear that all platforms under question are more
robust to male and female-centric clothing over non-binary ones. The absolute
percentage values for non-binary options in Table 3 are always lower than the
two binary genders which means that users are less inclined to pinpoint non-
binary choices for the recommendations that they see.

Delving deeper into the relevance of these recommendations, it is painfully
clear from Table 4 that even if non-binary recommendations are shown to the
users, these are not very relevant to the input, with Amazon’s recommendations
in particular receiving extremely low ratings. This brings to fore the issue
of how inclusive or engaging an LGBTQ+ person’s shopping experience would be
on such a platform. Amazon being one of the world’s largest e-commerce
platforms, needs to take responsibility for this and attempt to set a
benchmark that others may follow.

Using the recommendation ratings as a proxy for the ranking of the products,
we observe from Tables 6 and 6 that only the Lykdat platform has positive
correlation between the ground truth ranks and the participant ranks for non-
binary clothing items. Thus, not just are Amazon’s recommendations irrelevant,
they are also returned in a haphazard manner, thereby, creating a rather
unpleasant shopping experience for such shoppers. At the same time, Amazon has
the highest correlation for male-centric clothing items, which indicates that
the system as a whole isn’t broken, but is just not designed to handle non-
binary clothing inputs. Beagle.Vision has a similar performance as Amazon.

Finally, on comparing the male and the female participant responses, we
observe a very stark contrast in their perception of clothing as a whole.
Table 8 shows that male participants have a more binary opinion about clothing
items compared to females. Since buyer personas are one of the deciding
factors for inclusive design (CONTENTSQUARE, 2021), e-commerce platforms have
a lower incentive to serve to LGBTQ+ shoppers if the male participants are
assumed to be representative of the general shopper. This in turn puts off
LGBTQ+ shoppers from buying on these platforms and creates a cyclic system
that is doomed to only increase the divide.

##  6\. Final remarks

We hope that this case study will allow e-commerce platforms to take note of
the growing changes in society and be more inclusive in how they identify
clothing items and show recommendations. We would also like to see all
e-commerce platforms involved in this study to incorporate a gender-neutral or
non-binary clothing section in their catalogue so that it may serve as safe
shopping space for people who may face discrimination and persecution in
public for using physical shopping stores and/or changing rooms.

In future we would like to incorporate other platforms and scale up the study
in terms of more number of images and respondents. However, we believe that
all our observations shall remain the same.

## References

  * (1)
  * Amazon (2021a) Amazon. 2021a.  _Amazon StyleSnap_.  Retrieved September 23, 2021 from https://www.amazon.in/stylesnap
  * Amazon (2021b) Amazon. 2021b.  _Amazon.com_.  Retrieved September 23, 2021 from https://www.amazon.com
  * Beagle (2021) Beagle. 2021.  _Beagle Vision_.  Retrieved September 23, 2021 from https://beagle.vision/
  * CNBC (2021) CNBC. 2021.  _Lines between men’s and women’s fashion are blurring as more retailers embrace gender-fluid style_.  Retrieved October 08, 2021 from https://www.cnbc.com/2021/06/09/gender-fluid-fashion-booms-retailers-take-cues-from-streetwear-gen-z.html
  * CONTENTSQUARE (2021) CONTENTSQUARE. 2021\.  _Gender-Inclusivity and Retail: 4 Ways To Build a More Gender-Neutral Brand_.  Retrieved October 08, 2021 from https://contentsquare.com/blog/gender-inclusivity-and-retail/
  * Curatti.com (2021) Curatti.com. 2021\.  _The Importance of Visual Search in your eCommerce website_.  Retrieved September 23, 2021 from https://curatti.com/visual-search-ecommerce/
  * Dash et al. (2021) Abhisek Dash, Abhijnan Chakraborty, Saptarshi Ghosh, Animesh Mukherjee, and Krishna P Gummadi. 2021\.  When the Umpire is also a Player: Bias in Private Label Product Recommendations on E-commerce Marketplaces. In _Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency_. 873–884.
  * Devendorf et al. (2016) Laura Devendorf, Joanne Lo, Noura Howell, Jung Lin Lee, Nan-Wei Gong, M. Emre Karagozler, Shiho Fukuhara, Ivan Poupyrev, Eric Paulos, and Kimiko Ryokai. 2016\.  ”I Don’t Want to Wear a Screen”: Probing Perceptions of and Possibilities for Dynamic Displays on Clothing. In _Proceedings of the 2016 CHI Conference on Human Factors in Computing Systems_ _(CHI ’16)_. 6028–6039.
  * FIBRE2FASHION (2021) FIBRE2FASHION. 2021\.  _Gender-Free Clothing: Fashion Unbound_.  Retrieved October 08, 2021 from https://www.fibre2fashion.com/industry-article/8513/gender-free-clothing-fashion-unbound
  * Flipkart (2021) Flipkart. 2021.  _Flipkart.com_.  Retrieved September 23, 2021 from https://www.flipkart.com
  * Google (2021) Google. 2021.  _Google Lens_.  Retrieved September 23, 2021 from https://lens.google/
  * Kiapour et al. (2014) M. Hadi Kiapour, Kota Yamaguchi, Alexander C. Berg, and Tamara L. Berg. 2014.  Hipster Wars: Discovering Elements of Fashion Styles. In _Computer Vision – ECCV 2014_ , David Fleet, Tomas Pajdla, Bernt Schiele, and Tinne Tuytelaars (Eds.). Springer International Publishing, 472–488.
  * Lykdat (2021) Lykdat. 2021.  _Lykdat_.  Retrieved September 23, 2021 from https://lykdat.com/
  * Maity et al. (2019) Suman Kalyan Maity, Anshit E. Chaudhary, and Animesh Mukherjee. 2019.  ”Woman-Metal-White vs Man-Dress-Shorts”: Combining Social, Temporal and Image Signals to Understand Popularity of Pinterest Fashion Boards. In _Proceedings of the Thirteenth International Conference on Web and Social Media, ICWSM 2019, Munich, Germany, June 11-14, 2019_. 346–357.
  * Mehrabi et al. (2021) Ninareh Mehrabi, Fred Morstatter, Nripsuta Saxena, Kristina Lerman, and Aram Galstyan. 2021\.  A survey on bias and fairness in machine learning.  _ACM Computing Surveys (CSUR)_ 54, 6 (2021), 1–35.
  * Pan et al. (2012) Yue Pan, David Roedl, John C. Thomas, and Eli Blevis. 2012\.  Re-Conceptualizing Fashion in Sustainable HCI. In _Proceedings of the Designing Interactive Systems Conference_ _(DIS ’12)_. 621–630.
  * Park et al. (2016) Jaehyuk Park, Giovanni Luca Ciampaglia, and Emilio Ferrara. 2016.  Style in the Age of Instagram: Predicting Success within the Fashion Industry Using Social Media. In _Proceedings of the 19th ACM Conference on Computer-Supported Cooperative Work & Social Computing_ _(CSCW ’16)_. 64–73.
  * Pinterest (2021a) Pinterest. 2021a.  _Pinterest_.  Retrieved September 29, 2021 from https://www.pinterest.com/
  * Pinterest (2021b) Pinterest. 2021b.  _Pinterest Lens_.  Retrieved September 23, 2021 from https://help.pinterest.com/en/article/pinterest-lens
  * Ragot et al. (2020) Martin Ragot, Nicolas Martin, and Salomé Cojean. 2020\.  AI-Generated vs. Human Artworks. A Perception Bias Towards Artificial Intelligence?. In _Extended Abstracts of the 2020 CHI Conference on Human Factors in Computing Systems_ _(CHI EA ’20)_. 1–10.
  * Remotely (2021) Go Remotely. 2021\.  _28 Trendy Fashion Industry Statistics for a Luxurious 2021_.  Retrieved September 23, 2021 from https://goremotely.net/blog/fashion-industry-statistics/
  * Samanwaya Rautray (2018) Economic Times Samanwaya Rautray. 2018\.  _Section 377: Supreme Court rewrites history, homosexuality no longer a crime_.  Retrieved September 23, 2021 from https://economictimes.indiatimes.com/news/politics-and-nation/sc-delivers-historic-verdict-section-377-gone-being-gay-no-more-a-crime-in-india/articleshow/65696771.cms
  * Shopify.in (2021) Shopify.in. 2021\.  _What is Visual Search: How Retailers Can Use it to Enhance the Customer Experience_.  Retrieved September 23, 2021 from https://www.shopify.in/retail/what-is-visual-search
  * SmartInsights.com (2021) SmartInsights.com. 2021\.  _Third of consumers have used visual search_.  Retrieved October 08, 2021 from https://www.smartinsights.com/ecommerce/third-consumers-use-visual-search/
  * Snapchat (2021) Snapchat. 2021.  _Snapchat Scan_.  Retrieved September 23, 2021 from https://scan.snapchat.com/
  * Sun et al. (2021) Lingyun Sun, Jiaji Li, Danli Luo, Ziqi Fang, Yitao Fan, Zhi Yu, Yu Chen, Deying Pan, Yue Yang, Yijun Zhao, Jianzhe Gu, Lining Yao, Ye Tao, and Guanyun Wang. 2021.  Fashion Design with FlexTruss Approach. In _CHI ’21: CHI Conference on Human Factors in Computing Systems, Virtual Event / Yokohama Japan, May 8-13, 2021, Extended Abstracts_. 179:1–179:4.
  * Synced (2021) Synced. 2021.  _Visual Search Is Revolutionizing E-Commerce_.  Retrieved September 23, 2021 from https://medium.com/syncedreview/visual-search-is-revolutionizing-e-commerce-b27a37dbd296
  * Tepe (2021) Jan Tepe. 2021.  Investigating Sensory Perception as a Material for Fashion Design. In _CHI ’21: CHI Conference on Human Factors in Computing Systems, Virtual Event / Yokohama Japan, May 8-13, 2021, Extended Abstracts_. 24:1–24:10.
  * Vue.ai (2021) Vue.ai. 2021.  _Here’s How eCommerce Can Support Non-Binary Shoppers_.  Retrieved October 08, 2021 from https://vue.ai/blog/ai-in-retail/non-binary-ecommerce-fashion-shopping/
  * Weinhardt et al. (2017) Lance S Weinhardt, Patricia Stevens, Hui Xie, Linda M Wesp, Steven A John, Immaculate Apchemengich, David Kioko, Shannon Chavez-Korell, Katherine M Cochran, Jennifer M Watjen, et al. 2017\.  Transgender and gender nonconforming youths’ public facilities use and psychological well-being: a mixed-method study.  _Transgender health_ 2, 1 (2017), 140–150.
  * Yamaguchi et al. (2014) Kota Yamaguchi, Tamara L. Berg, and Luis E. Ortiz. 2014\.  Chic or Social: Visual Popularity Analysis in Online Fashion Networks. In _Proceedings of the 22nd ACM International Conference on Multimedia_ _(MM ’14)_. 773–776.
  * YouthKiAwaaz (2021) YouthKiAwaaz. 2021\.  _Dear E-Commerce Platforms, Your Diversity Policies Can’t Ignore The Customers!_ Retrieved October 08, 2021 from https://www.youthkiawaaz.com/2020/09/online-shopping-where-is-diversity-where-is-inclusion/


# Groups whose Chermak-Delgado lattice is a subgroup lattice of an elementary
abelian $p$-group††thanks: This work was supported by NSFC (No. 11971280
&11771258)

Lijian An
Department of Mathematics, Shanxi Normal University
Linfen, Shanxi 041004, P. R. China

###### Abstract

The Chermak-Delgado lattice of a finite group $G$ is a self-dual sublattice of
the subgroup lattice of $G$. In this paper, we focus on finite groups whose
Chermak-Delgado lattice is a subgroup lattice of an elementary abelian
$p$-group. We prove that such groups are nilpotent of class $2$. We also prove
that, for any elementary abelian $p$-group $E$, there exists a finite group
$G$ such that the Chermak-Delgado lattice of $G$ is a subgroup lattice of $E$.

Keywords Chermak-Delgado lattice quasi-antichain special $p$-groups

2000 Mathematics subject classification: 20D15 20D30.

##  1 Introduction

Suppose that $G$ is a finite group, and $H$ is a subgroup of $G$. The Chermak-
Delgado measure of $H$ (in $G$) is denoted by $m_{G}(H)$, and defined as
$m_{G}(H)=|H|\cdot|C_{G}(H)|.$ The maximal Chermak-Delgado measure of $G$ is
denoted by $m^{*}(G)$, and defined as

$m^{*}(G)=\max\\{m_{G}(H)\mid H\leq G\\}.$

Let

$\mathcal{CD}(G)=\\{H\mid m_{G}(H)=m^{*}(G)\\}.$

Then the set $\mathcal{CD}(G)$ forms a sublattice of the subgroup lattice of
$G$, which is called the Chermak-Delgado lattice of $G$. It was first
introduced by Chermak and Delgado [7], and revisited by Isaacs [11]. In the
last years, there has been a growing interest in understanding this lattice
(see e.g. [1-6], [8-9], [12-18]).

Notice that a Chermak-Delgado lattice is always self-dual. It is natural to
ask the question: which types of self-dual lattices can be as Chermak-Delgado
lattices of finite groups. Some special cases of this question are proposed
and solved. In [3], it is proved that, for any integer $n$, a chain of length
$n$ can be a Chermak-Delgado lattice of a finite $p$-group.

A quasi-antichain is a lattice consisting of a maximum, a minimum, and the
atoms of the lattice. The width of a quasi-antichain is the number of atoms.
For a positive integer $w\geq 3$, a quasi-antichain of width $w$ is denoted by
$\mathcal{M}_{w}$. In [4], it was proved that $\mathcal{M}_{w}$ can be as a
Chermak-Delgado lattice of a finite group if and only if $w=1+p^{a}$ for some
positive integer $a$ and some prime $p$. The following theorem gives more
self-dual lattices which can be as Chermak-Delgado lattices of finite groups.

######  Theorem 1.1.

([2]) If $\mathcal{L}$ is a Chermak-Delgado lattice of a finite $p$-group $G$
such that both $G/Z(G)$ and $G^{\prime}$ are elementary abelian, then are
$\mathcal{L}^{+}$ and $\mathcal{L}^{++}$, where $\mathcal{L}^{+}$ is a mixed
$3$-string with center component isomorphic to $\mathcal{L}$ and the remaining
components being $m$-diamonds (a lattice with subgroups in the configuration
of an $m$-dimensional cube), $\mathcal{L}^{++}$ is a mixed $3$-string with
center component isomorphic to $\mathcal{L}$ and the remaining components
being lattice isomorphic to $\mathcal{M}_{p+1}$.

For a finite group $G$, we use $\mathcal{L}(G)$ to denoted the subgroup
lattice of $G$. We use $E_{p^{n}}$ to denote the elementary abelian $p$-group
of order $p^{n}$. It is well-known that $\mathcal{L}(E_{p^{n}})$ is self-dual.
Let $G$ be an extra-special $p$-group of order $p^{2n+1}$. Then
$\mathcal{CD}(G)$ is isomorphic to $\mathcal{L}(E_{p^{2n}})$ (see [9, Example
2.8]). In this paper, we focus on finite groups whose Chermak-Delgado lattice
is isomorphic to $\mathcal{L}(E_{p^{n}})$. The main results are:

Theorem A. Let $G$ be a finite group with $G\in\mathcal{CD}(G)$. Suppose that
$\mathcal{CD}(G)$ is isomorphic to $\mathcal{L}(E_{p^{n}})$, where $n\geq 2$.
Then $G=P\times Q$, where $P$ is the Sylow $p$-subgroup of $G$ such that
$P/Z(P)$ is elementary abelian, $Q$ is the abelian Hall $p^{\prime}$-subgroup
of $G$. Moreover, $\mathcal{CD}(G)\cong\mathcal{CD}(P)$ as lattice.

Theorem B. For any integer $n$ and a prime $p$, there exists a special
$p$-group $G$ such that $\mathcal{CD}(G)$ is isomorphic to
$\mathcal{L}(E_{p^{n}})$.

For a Chermak-Delgado lattice, the following properties is basic and is often
used in this paper. We will not point out when we use them.

######  Theorem 1.2.

[7] Suppose that $G$ is a finite group and $H,K\in\mathcal{CD}(G)$.

$\langle H,K\rangle=HK$. Hence a Chermak-Delgado lattice is modular.

$C_{G}(H\cap K)=C_{G}(H)C_{G}(K)$.

$C_{G}(H)\in\mathcal{CD}(G)$ and $C_{G}(C_{G}(H))=H$. Hence a Chermak-Delgado
lattice is self-dual.

Let $M$ be the maximal member of $\mathcal{CD}(G)$. Then $M$ is characteristic
in $G$ and $\mathcal{CD}(M)=\mathcal{CD}(G)$.

The minimal member of $\mathcal{CD}(G)$ is characteristic, abelian, and
contains $Z(G)$.

##  2 Quasi-antichain intervals in Chermak-Delgado lattices

If $n\geq 2$, then every interval of length $2$ in $\mathcal{L}(E_{p^{n}})$ is
a quasi-antichain of width $p+1$. Hence we start our argument from
investigating quasi-antichain intervals in Chermak-Delgado lattices. Following
[4], we use $[\hskip-5.0pt[\hskip 2.0ptL,H\hskip 2.0pt]\hskip-5.0pt]\hskip
2.0pt$ to denote the interval from $L$ to $H$ in $\mathcal{CD}(G)$.

######  Lemma 2.1.

([4, Proposition 2 & Theorem 4]) Let $G$ be a finite group with an interval
$[\hskip-5.0pt[\hskip 2.0ptL,H\hskip 2.0pt]\hskip-5.0pt]\hskip
2.0pt\cong\mathcal{M}_{w}$ in $\mathcal{CD}(G)$, where $w\geq 3$, and $K$ be
an atom of the quasi-antichain. Then $K\trianglelefteq H$, $L\trianglelefteq
H$, and there exists a prime $p$ and positive integers $a,b$ with $b\leq a$
such that $H/L$ is elementary abelian $p$-groups of order $p^{2a}$,
$|H/K|=|K/L|=p^{a}$ and $w=p^{b}+1$.

######  Lemma 2.2.

Let $G$ be a finite group with an interval $[\hskip-5.0pt[\hskip
2.0ptL,H\hskip 2.0pt]\hskip-5.0pt]\hskip 2.0pt$ of length $l$ in
$\mathcal{CD}(G)$. Suppose that every interval of length $2$ in
$[\hskip-5.0pt[\hskip 2.0ptL,H\hskip 2.0pt]\hskip-5.0pt]\hskip 2.0pt$ is a
quasi-antichain of width $\geq 3$. Then $L\trianglelefteq H$, and there exists
a prime $p$ and a positive integer $a$ such that $H/L$ is elementary abelian
$p$-groups of order $p^{al}$. Moreover, if $[\hskip-5.0pt[\hskip
2.0ptL_{1},H_{1}\hskip 2.0pt]\hskip-5.0pt]\hskip 2.0pt$ is an interval of
length $2$ in $[\hskip-5.0pt[\hskip 2.0ptL,H\hskip 2.0pt]\hskip-5.0pt]\hskip
2.0pt$, then the width of $[\hskip-5.0pt[\hskip 2.0ptL_{1},H_{1}\hskip
2.0pt]\hskip-5.0pt]\hskip 2.0pt$ is $p^{b}+1$ for some integer $b$.

Proof We use $l_{H/L}$ to denote the length of $[\hskip-5.0pt[\hskip
2.0ptL,H\hskip 2.0pt]\hskip-5.0pt]\hskip 2.0pt$. If $l=l_{H/L}=2$, then the
conclusions follow from Lemma 2.1. In the following, we may assume that $l\geq
3$. Let $L=J_{0}<J_{1}<J_{2}<\dots<J_{l}=H$ be a maximal chain.

For $0\leq i\leq l-2$, $[\hskip-5.0pt[\hskip 2.0ptJ_{i},J_{i+2}\hskip
2.0pt]\hskip-5.0pt]\hskip 2.0pt$ is a quasi-antichain of width $\geq 3$. By
Lemma 2.1, there exist primes $p_{i}$ and positive integers $a_{i}$ such that,
$|J_{i+2}/J_{i+1}|=|J_{i+1}/J_{i}|=p_{i}^{a_{i}}$. It is easy to see that
these $p_{i}$ are coincide to a prime $p$ and these $a_{i}$ are coincide to an
integer $a$. Hence $|H:L|=p^{al}$.

Since $[\hskip-5.0pt[\hskip 2.0ptL,J_{2}\hskip 2.0pt]\hskip-5.0pt]\hskip
2.0pt$ is a quasi-antichain of width $\geq 3$, there is an atom
$L<K_{1}<J_{2}$ such that $K_{1}\neq J_{1}$. Since
$l_{H/J_{1}}=l_{H/K_{1}}=l-1$, by induction, $J_{1}\trianglelefteq H$,
$K_{1}\trianglelefteq H$ and $H/J_{1}$ and $H/K_{1}$ are elementary
$p$-groups. It follows that $L=J_{1}\cap K_{1}\unlhd H$ and $H/L\lesssim
H/J_{1}\times H/K_{1}$ is an elementary abelian $p$-group of order $p^{al}$.

If $[\hskip-5.0pt[\hskip 2.0ptL_{1},H_{1}\hskip 2.0pt]\hskip-5.0pt]\hskip
2.0pt$ is an interval of length $2$ in $[\hskip-5.0pt[\hskip 2.0ptL,H\hskip
2.0pt]\hskip-5.0pt]\hskip 2.0pt$, then, by Lemma 2.1, the width of
$[\hskip-5.0pt[\hskip 2.0ptL_{1},H_{1}\hskip 2.0pt]\hskip-5.0pt]\hskip 2.0pt$
is $p^{b}+1$ for some integer $b$. $\Box$

######  Theorem 2.3.

Let $G$ be a finite group with $G\in\mathcal{CD}(G)$. Suppose that every
interval of length $2$ in $\mathcal{CD}(G)$ is a quasi-antichain of width
$\geq 3$. Then there exists a prime $p$ such that the width of an interval of
length $2$ in $\mathcal{CD}(G)$ is $p^{b}+1$ for some integer $b$. Moreover,
$G=P\times Q$, where $P$ is the Sylow $p$-subgroup of $G$, nilpotent of class
$2$, $Q$ is the abelian Hall $p^{\prime}$-subgroup of $G$, and
$\mathcal{CD}(G)\cong\mathcal{CD}(P)$ as lattice.

Proof By Lemma 2.2, $G/Z(G)$ ia an elementary abelian $p$-group for some prime
$p$ and the width of an interval of length $2$ in $\mathcal{CD}(G)$ is
$p^{b}+1$ for some integer $b$. Hence $G$ is nilpotent. Let $P$ be the Sylow
$p$-subgroup and $Q$ the Hall $p^{\prime}$-subgroup. Then $G=P\times Q$.
Obviously, $Q\in Z(G)$. Therefore
$\mathcal{CD}(G)\cong\mathcal{CD}(P)\times\mathcal{CD}(Q)\cong\mathcal{CD}(P)$.
$\Box$

Proof of Theorem A. Since $\mathcal{CD}(G)$ is isomorphic to
$\mathcal{L}(E_{p^{n}})$, where $n\geq 2$, every interval of length $2$ in
$\mathcal{CD}(G)$ is a quasi-antichain of width $p+1$. By Theorem 2.3, there
exists a prime $p_{1}$ such that the width of an interval of length $2$ in
$\mathcal{CD}(G)$ is $p_{1}^{b}+1$ for some integer $b$. Hence $p_{1}=p$ and
$b=1$. Other results hold obviously. $\Box$

##  3 Proof of Theorem B

######  Lemma 3.1.

[11, Lemma 1.43] Suppose that $G$ is finite group. If $H,K\leq G$, then

$m_{G}(H)\cdot m_{G}(K)\leq m_{G}(\langle H,K\rangle)\cdot m_{G}(H\cap K).$

Moreover, equality occurs if and only if $\langle H,K\rangle=HK$ and
$C_{G}(H\cap K)=C_{G}(H)C_{G}(K)$.

######  Lemma 3.2.

Suppose that $G$ is finite group. If $K\leq H\leq G$, then

$\frac{m_{H}(K)}{m_{G}(K)}\leq\frac{m_{H}(H)}{m_{G}(H)}.$

Moreover, equality occurs if and only if $C_{G}(K)\leq HC_{G}(H)$.

Proof By calculation,

$\frac{m_{H}(K)}{m_{G}(K)}=\frac{|K|\cdot|C_{H}(K)|}{|K|\cdot|C_{G}(K)|}=\frac{|C_{H}(K)|}{|C_{G}(K)|}=\frac{|H\cap
C_{G}(K)|}{|C_{G}(K)|}=\frac{|H|}{|HC_{G}(K)|}$

and

$\frac{m_{H}(H)}{m_{G}(H)}=\frac{|H|\cdot|C_{H}(H)|}{|H|\cdot|C_{G}(H)|}=\frac{|C_{H}(H)|}{|C_{G}(H)|}=\frac{|H\cap
C_{G}(H)|}{|C_{G}(H)|}=\frac{|H|}{|HC_{G}(H)|}.$

Since $K\leq H$, $C_{G}(H)\leq C_{G}(K)$. Hence $|HC_{G}(H)|\leq|HC_{G}(K)|$,
where equality occurs if and only if $C_{G}(K)\leq HC_{G}(H)$. Thus

$\frac{m_{H}(K)}{m_{G}(K)}=\frac{m_{H}(H)}{m_{G}(H)}\frac{|HC_{G}(H)|}{|HC_{G}(K)|}\leq\frac{m_{H}(H)}{m_{G}(H)},$

where equality occurs if and only if $C_{G}(K)\leq HC_{G}(H)$. $\Box$

######  Lemma 3.3.

Suppose that $G$ is finite group, $H\leq G$ such that $G=HC_{G}(H)$. If
$H\in\mathcal{CD}(H)$, then $H$ contains in the unique maximal member of
$\mathcal{CD}(G)$.

Proof Let $M$ be the unique maximal member of $\mathcal{CD}(G)$. Since
$C_{G}(M\cap H)\leq G=HC_{G}(H)$, by Lemma 3.2,

$\frac{m_{H}(M\cap H)}{m_{G}(M\cap H)}=\frac{m_{H}(H)}{m_{G}(H)}.$

Since $H\in\mathcal{CD}(H)$, $m_{H}(M\cap H)\leq m_{H}(H)$. It follows that
$m_{G}(M\cap H)\leq m_{G}(H).$ By Lemma 3.1,

$m_{G}(H)\cdot m_{G}(M)\leq m_{G}(\langle H,M\rangle)\cdot m_{G}(M\cap H).$

It follows that $m_{G}(\langle H,M\rangle)\geq m_{G}(M)=m^{*}(G)$. Hence
$\langle H,M\rangle\in\mathcal{CD}(G)$. Since $M$ is maximal, $H\leq M$.
$\Box$

######  Theorem 3.4.

Suppose $H\in\mathcal{CD}(G)$. Then $\mathcal{CD}(H)$ is just the interval
$[\hskip-5.0pt[\hskip 2.0ptZ(H),H\hskip 2.0pt]\hskip-5.0pt]\hskip 2.0pt$ in
$\mathcal{CD}(G)$.

Proof Since $H\in\mathcal{CD}(G)$, $m_{G}(H)=m^{*}(G)$. By Lemma 3.2,

$m_{H}(H)\geq
m_{G}(H)\frac{m_{H}(K)}{m_{G}(K)}=m^{*}(G)\frac{m_{H}(K)}{m_{G}(K)}\geq
m_{H}(K)$ (1)

for all $K\leq H$. It follows that $H\in\mathcal{CD}(H)$ and
$m^{*}(H)=m_{H}(H)$. Moreover, “=” holds in Equation (1) if and only if
$C_{G}(K)\leq HC_{G}(H)$ and $m_{G}(K)=m^{*}(G)$. Notice that $C_{G}(K)\leq
HC_{G}(H)$ if and only if $K\geq C_{G}(HC_{G}(H))=H\cap C_{G}(H)=Z(H)$, and
$m_{G}(K)=m^{*}(G)$ if and only if $K\in\mathcal{CD}(G)$.
$K\in\mathcal{CD}(H)$ if and only if

“=” holds in Equation (1)

if and only if $K\in[\hskip-5.0pt[\hskip 2.0ptZ(H),H\hskip
2.0pt]\hskip-5.0pt]\hskip 2.0pt$. Hence $\mathcal{CD}(H)$ is just the interval
$[\hskip-5.0pt[\hskip 2.0ptZ(H),H\hskip 2.0pt]\hskip-5.0pt]\hskip 2.0pt$.
$\Box$

######  Lemma 3.5.

Let $\mathbb{F}$ be a field and $A,B$ be $n\times n$ matrices, where $n\geq
3$. If $AZ=ZB$ for all anti-symmetric matrix $Z$, then $A,B$ are scalar
matrices.

Proof Let $A=(a_{ij})$ and $B=(b_{ij})$. Since
$A(E_{ij}-E_{ji})=(E_{ij}-E_{ji})B$ for $i\neq j$, by calculation, we have
$a_{ii}=b_{jj}$ and

$a_{ki}=a_{kj}=b_{ik}=b_{jk}=0$, if $k\neq i,j.$

Since $n\geq 3$, all $a_{ki}$ and $b_{ki}$ are zero if $k\neq i$, and all
$a_{ii}$ and $b_{ii}$ are coincide. Hence $A$ and $B$ are scalar matrices.
$\Box$

######  Construction 3.6.

For a prime $p$, let $P=\langle x,y,w;z_{1},z_{2},z_{3}\mid
x^{p}=y^{p}=w^{p}=1,[x,y]=z_{1},[y,w]=z_{2},[w,x]=z_{3},z_{i}^{p}=[z_{i},x]=[z_{i},y]=[z_{i},w]=1\
\mbox{where}\ i=1,2,3\rangle$. Then it is easy to check that
$\Phi(P)=Z(P)=P^{\prime}=\langle z_{1},z_{2},z_{3}\rangle$ is of order
$p^{3}$, $|P|=p^{6}$, and $\mathcal{CD}(P)=\\{P,Z(P)\\}$.

Let $G_{n}$ be the group which is the central product of $n$ copies of $P$.
Thus $G_{n}$ has order $p^{3n+3}$ and is generated by $3n$ elements of order
$p$, $x_{1},\dots,x_{n},y_{1},\dots,y_{n},w_{1},\dots,w_{n}$ subject to the
defining relations:

$[x_{i},x_{j}]=[y_{i},y_{j}]=[w_{i},w_{j}]=[x_{i},y_{j}]=[y_{i},w_{j}]=[w_{i},x_{j}]=1$
if $i\neq j$,

$[x_{i},y_{i}]=z_{1}$, $[y_{i},z_{i}]=z_{2}$, $[w_{i},x_{i}]=z_{3}$,

$[z_{j},x_{i}]=[z_{j},y_{i}]=[z_{j},w_{i}]=1$.

It is easy to Check that $G_{n}$ is a special $p$-group with
$\Phi(G_{n})=Z(G_{n})=G_{n}^{\prime}=\langle z_{1},z_{2},z_{3}\rangle$.

######  Theorem 3.7.

Let $G=G_{n}$ which is defined in Construction 3.6. Let $P_{i}=\langle
x_{i},y_{i},w_{i}\rangle$, where $1\leq i\leq n$. Then

$G\in\mathcal{CD}(G)$, $m^{*}(G)=p^{3n+6}$, and $P_{i}\in\mathcal{CD}(G)$.

If $H\in\mathcal{CD}(G)$, then $|H|=p^{3m+3}$ for some $0\leq m\leq n$.

We use $F_{p}$ to denote the finite field $\mathbb{Z}/p\mathbb{Z}$. For a
vector

$v=(s_{1},s_{2},\dots,s_{n})$

of $F_{p}^{n}$, we use $v^{\varphi}$ to denote the subgroup
$\langle\alpha_{v},\beta_{v},\gamma_{v},Z(G)\rangle$, where

$\alpha_{v}=\prod_{i=1}^{n}x_{i}^{s_{i}},\
\beta_{v}=\prod_{i=1}^{n}y_{i}^{s_{i}},\
\gamma_{v}=\prod_{i=1}^{n}w_{i}^{s_{i}}.$

Let $\tilde{v}=(t_{1},t_{2},\dots,t_{n})$. Define an inner product on
$F_{p}^{n}$ with $\langle v,\tilde{v}\rangle=\sum_{i=1}^{n}s_{i}t_{i}$. Then
$[v^{\varphi},{\tilde{v}}^{\varphi}]=1$ if and only if $\langle
v,\tilde{v}\rangle=0.$

Suppose that $U$ is an $m$-dimensional subspace of $F_{p}^{n}$. We use
$U^{\varphi}$ to denote the subgroup $\prod_{u\in U}{u}^{\varphi}$ of $G$.
Then $|U^{\varphi}|=p^{3m+3}$.

$U^{\varphi}\in\mathcal{CD}(G)$. Moreover, let

$U^{\perp}=\\{v\in F_{p}^{n}\mid\langle u,v\rangle=0\ \mbox{for all}\ u\in U\
\\}.$

Then $(U^{\perp})^{\varphi}=C_{G}(U^{\varphi})$.

If $H\in\mathcal{CD}(G)$, then there exists a subspace $U$ of $F_{p}^{n}$ such
that $H=U^{\varphi}$.

Proof (1) It is easy to see that $C_{G}(P_{i})=\prod_{j\neq i}P_{j}\cong
G_{n-1}$ for $1\leq i\leq n$. Hence $m_{G}(G)=m_{G}(P_{i})=p^{3n+6}$. Since
$\mathcal{CD}(P_{i})=\\{P_{i},Z(G)\\}$ and $G=P_{i}C_{G}(P_{i})$, by Lemma
3.3, $P_{i}$ contains in the unique maximal member of $\mathcal{CD}(G)$. Hence
$G$ is the unique maximal member of $\mathcal{CD}(G)$,
$m^{*}(G)=m_{G}(G)=p^{3n+6}$, and $P_{i}\in\mathcal{CD}(G)$.

(2) It is trivial for $n=1$. Assume that $n\geq 2$. Let
$Q_{n}=C_{G}(P_{n})=\prod_{i=1}^{n-1}P_{i}\cong G_{n-1}$. If $P_{n}\leq H$,
then $C_{G}(H)\leq Q_{n}$. By Theorem 3.4, $C_{G}(H)\in\mathcal{CD}(Q_{n})$.
By induction, $|C_{G}(H)|=p^{3m+3}$ for some $0\leq m\leq n-1$. Hence
$|H|=m^{*}(G)/|C_{G}(H)|=p^{3(n-m)+3}$, where $1\leq n-m\leq n$. If
$P_{n}\not\leq H$, then, by above argument, $|HP_{n}|=p^{3m+3}$ for some
$1\leq m\leq n$. By Theorem 3.4, $H\cap P_{n}\in\mathcal{CD}(P_{n})$. Since
$\mathcal{CD}(P_{n})=\\{P_{n},Z(G)\\}$ and $P_{n}\not\leq H$, $H\cap
P_{n}=Z(G)$. Hence

$|H|=\frac{|HP_{n}|\cdot|H\cap P_{n}|}{|P_{n}|}=p^{3(m-1)+3}.$

(3) By calculation,

$[\alpha_{v},\beta_{\tilde{v}}]=[\alpha_{\tilde{v}},\beta_{v}]=z_{1}^{\langle
v,\tilde{v}\rangle},$
$[\beta_{v},\gamma_{\tilde{v}}]=[\beta_{\tilde{v}},\gamma_{{v}}]=z_{2}^{\langle
v,\tilde{v}\rangle},$
$[\gamma_{v},\alpha_{\tilde{v}}]=[\gamma_{\tilde{v}},\alpha_{{v}}]=z_{3}^{\langle
v,\tilde{v}\rangle}.$

Since
$[\alpha_{v},\alpha_{\tilde{v}}]=[\beta_{v},\beta_{\tilde{v}}]=[\gamma_{v},\gamma_{\tilde{v}}]=1$,
$[v^{\varphi},{\tilde{v}}^{\varphi}]=1$ if and only if $\langle
v,\tilde{v}\rangle=0.$

(4) Let

$A=\langle\alpha_{u}\mid u\in U\rangle Z(G),$

$B=\langle\beta_{u}\mid u\in U\rangle Z(G)$,

$C=\langle\gamma_{u}\mid u\in U\rangle Z(G).$

Then $|A/Z(G)|=|B/Z(G)|=|C/Z(G)|=p^{m}$. Since

$U^{\varphi}/Z(G)=A/Z(G)\times B/Z(G)\times C/Z(G),$

$|U^{\varphi}/Z(G)|=p^{3m}$. Hence $|U^{\varphi}|=p^{3m+3}$.

(5) Notice that $U^{\perp}$ is an $(n-m)$-dimensional subspace of $F_{p}^{n}$.
By (4), $|({U^{\perp}})^{\varphi}|=p^{3(n-m)+3}$. By (3),
$[({U^{\perp}})^{\varphi},U^{\varphi}]=1$. Hence

$m_{G}(U^{\varphi})=|U^{\varphi}|\cdot|C_{G}(U^{\varphi})|\geqslant|U^{\varphi}|\cdot|({U^{\perp}})^{\varphi}|=p^{3n+6}=m^{*}(G).$

It follows that $U^{\varphi}\in\mathcal{CD}(G)$ and
$C_{G}(U^{\varphi})=({U^{\perp}})^{\varphi}.$

(6) If $n=1$, then the conclusion is trivial. In the following, we assume that
$n\geq 2$. Let $Q_{n}=C_{G}(P_{n})=\prod_{i=1}^{n-1}P_{i}\cong G_{n-1}$.

Case 1. $H\leq Q_{n}$.

By Theorem 3.4, $H\in\mathcal{CD}(Q_{n})$. By induction, there exists a
subspace $U$ of $F_{p}^{n-1}\times\\{0\\}\subset F_{p}^{n}$ such that
$H=U^{\varphi}$.

Case 2. $P_{n}\leq H$.

Notice that $C_{G}(H)\leq C_{G}(P_{n})=Q_{n}$. By Case 1, there exists a
subspace $U$ of $F_{p}^{n-1}\times\\{0\\}\subset F_{p}^{n}$ such that
$C_{G}(H)=U^{\varphi}$. By (5), $H=C_{G}(C_{G}(H))=({U^{\perp}})^{\varphi}$.

Case 3. $H\not\leq Q_{n}$ and $P_{n}\not\leq H$.

By (2), $|H|=p^{3m+3}$ for some $1\leq m\leq n-1$. Also by (2),
$|HQ_{n}|=p^{3m^{\prime}+3}$ for some $n-1\leq m^{\prime}\leq n$. Since
$HQ_{n}>Q_{n}$, $m^{\prime}=n$ and hence $HQ_{n}=G$.

Let $H_{1}=HP_{n}$, $H_{2}=HP_{n}\cap Q_{n}$ and $H_{3}=H\cap Q_{n}$. Since
$H\cap P_{n}\in[\hskip-5.0pt[\hskip 2.0ptZ(P_{n}),P_{n}\hskip
2.0pt]\hskip-5.0pt]\hskip 2.0pt$, by Theorem 3.4, $H\cap
P_{n}\in\mathcal{CD}(P_{n})=\\{P_{n},Z(G)\\}$. It follows that $H\cap
P_{n}=Z(G)$. Hence

$|H_{1}|=|HP_{n}|=\frac{|H|\cdot|P_{n}|}{|H\cap P_{n}|}=p^{3(m+1)+3}.$

Since $H_{1}Q_{n}=HQ_{n}=G$,

$|H_{2}|=|H_{1}\cap Q_{n}|=\frac{|H_{1}|\cdot|Q_{n}|}{|H_{1}Q_{n}|}=p^{3m+3}$

and

$|H_{3}|=|H\cap Q_{n}|=\frac{|H|\cdot|Q_{n}|}{|HQ_{n}|}=p^{3(m-1)+3}.$

By Case 1, there exist an $m$-dimensional subspace $U_{2}$ and an
$(m-1)$-dimensional subspace $U_{3}$ of $F_{p}^{n}\times\\{0\\}\subset
F_{p}^{n}$ such that $H_{2}=U_{2}^{\varphi}$ and $H_{3}=U_{3}^{\varphi}$.
Since $H_{3}\leq H_{2}$, $U_{3}\leq U_{2}$. Hence there exists a vector $u\in
F_{p}^{n}\times\\{0\\}\subset F_{p}^{n}$ such that $H_{2}=H_{3}u^{\varphi}$.
Let $H^{*}=H\cap u^{\varphi}P_{n}$. Then

$H=H\cap HP_{n}=H\cap H_{2}P_{n}=H\cap H_{3}u^{\varphi}P_{n}=H_{3}(H\cap
u^{\varphi}P_{n})=H_{3}H^{*}.$

Since $H^{*}P_{n}=(H\cap u^{\varphi}P_{n})P_{n}=HP_{n}\cap
u^{\varphi}P_{n}=u^{\varphi}P_{n}$, we may assume that
$H^{*}=\langle\alpha,\beta,\gamma,Z(G)\rangle$ where

$\alpha=\alpha_{u}x_{n}^{a_{11}}y_{n}^{a_{12}}w_{n}^{a_{13}},\
\beta=\beta_{u}x_{n}^{a_{21}}y_{n}^{a_{22}}w_{n}^{a_{23}},\
\gamma=\gamma_{u}x_{n}^{a_{31}}y_{n}^{a_{32}}w_{n}^{a_{33}}.$

Since $H_{2}=H_{3}u^{\varphi}$,

$Hu^{\varphi}=HH_{3}u^{\varphi}=HH_{2}=H(HP_{n}\cap Q_{n})=HP_{n}\cap
HQ_{n}=HP_{n}.$

Hence

$H^{*}u^{\varphi}=(H\cap u^{\varphi}P_{n})u^{\varphi}=Hu^{\varphi}\cap
u^{\varphi}P_{n}=u^{\varphi}P_{n}.$

It follows that the matrix $(a_{ij})_{3\times 3}$ is invertible.

Let $K=C_{G}(H)$. Then $P_{n}\not\leq K$ and $K\not\leq Q_{n}$. Similarly,
there exist a vector $v\in F_{p}^{n}\times\\{0\\}\subset F_{p}^{n}$ such that
$K=(K\cap Q_{n})(K\cap v^{\varphi}P_{n})$, and we may assume that $K\cap
v^{\varphi}P_{n}=\langle\tilde{\alpha},\tilde{\beta},\tilde{\gamma},Z(G)\rangle$
where

$\tilde{\alpha}=\alpha_{v}x_{n}^{b_{11}}y_{n}^{b_{12}}w_{n}^{b_{13}},\
\tilde{\beta}=\beta_{u}x_{n}^{b_{21}}y_{n}^{b_{22}}w_{n}^{b_{23}},\
\tilde{\gamma}=\gamma_{u}x_{n}^{b_{31}}y_{n}^{b_{32}}w_{n}^{b_{33}}.$

The matrix $(b_{ij})_{3\times 3}$ is also invertible.

For convenience, in the following, the operation of the group $G$ is written
as addition. Hence

$\left(\begin{array}[]{c}\alpha\\\ \beta\\\ \gamma\\\
\end{array}\right)=\left(\begin{array}[]{c}\alpha_{u}\\\ \beta_{u}\\\
\gamma_{u}\\\ \end{array}\right)+A\left(\begin{array}[]{c}x_{n}\\\ y_{n}\\\
z_{n}\\\ \end{array}\right)$

where $A=(a_{ij})_{3\times 3}$, and
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\alpha_{v},\beta_{v},\gamma_{v})+(x_{n},y_{n},z_{n})B^{T}$
where $B=(b_{ij})_{3\times 3}.$ Notice that
$[u^{\varphi},P_{n}]=[v^{\varphi},P_{n}]=0$. By calculation, we have

$\left(\begin{array}[]{ccc}[\alpha,\tilde{\alpha}]&[\alpha,\tilde{\beta}]&[\alpha,\tilde{\gamma}]\\\
\mbox{[}\beta,\tilde{\alpha}\mbox{]}&[\beta,\tilde{\beta}]&[\beta,\tilde{\gamma}]\\\
\mbox{[}\gamma,\tilde{\alpha}\mbox{]}&[\gamma,\tilde{\beta}]&[\gamma,\tilde{\gamma}]\\\
\end{array}\right)=\left(\begin{array}[]{ccc}[\alpha_{u},\alpha_{v}]&[\alpha_{u},\beta_{v}]&[\alpha_{u},\gamma_{v}]\\\
\mbox{[}\beta_{u},\alpha_{v}\mbox{]}&[\beta_{u},\beta_{v}]&[\beta_{u},\gamma_{v}]\\\
\mbox{[}\gamma_{u},\alpha_{v}\mbox{]}&[\gamma_{u},\beta_{v}]&[\gamma_{u},\gamma_{v}]\\\
\end{array}\right)+AZB^{T}$ (2)

where

$Z=\left(\begin{array}[]{ccc}[x_{n},x_{n}]&[x_{n},y_{n}]&[x_{n},w_{n}]\\\
\mbox{[}y_{n},x_{n}\mbox{]}&[y_{n},y_{n}]&[y_{n},w_{n}]\\\
\mbox{[}w_{n},x_{n}\mbox{]}&[w_{n},y_{n}]&[w_{n},w_{n}]\\\
\end{array}\right)=\left(\begin{array}[]{ccc}0&z_{1}&z_{2}\\\
-z_{1}&0&-z_{3}\\\ z_{3}&-z_{2}&0\\\ \end{array}\right).$

Since $[H^{*},K\cap v^{\varphi}P_{n}]=0$, the left of Equation (2) equals to
$O_{3\times 3}$. Since

$\left(\begin{array}[]{ccc}[\alpha_{u},\alpha_{v}]&[\alpha_{u},\beta_{v}]&[\alpha_{u},\gamma_{v}]\\\
\mbox{[}\beta_{u},\alpha_{v}\mbox{]}&[\beta_{u},\beta_{v}]&[\beta_{u},\gamma_{v}]\\\
\mbox{[}w_{n},x_{n}\mbox{]}&[\gamma_{u},\beta_{v}]&[\gamma_{u},\gamma_{v}]\\\
\end{array}\right)=\langle u,v\rangle Z,$

we have

$AZB^{T}=-\langle u,v\rangle Z$ (3)

Since $A$ is invertible, we have $ZB^{T}=-\langle u,v\rangle A^{-1}Z$. By
Lemma 3.5, $A$ is a scalar matrix. Let $u^{*}=u+(0,\cdots,0,a_{11})$. Then
$H^{*}=(u^{*})^{\varphi}$. Let $U=U_{3}+\hbox{\rm span}(u^{*})$. Then
$H=H_{3}H^{*}=U^{\varphi}$. $\Box$

Proof of Theorem B. Let $G=G_{n}$ which is defined in Construction 3.6. By
Theorem 3.7, $\mathcal{CD}(G)$ is isomorphic to the subspace lattice of
$F_{p}^{n}$, which is isomorphic to $\mathcal{L}(E_{p^{n}})$. $\Box$

## References

  * [1] L. An, Groups whose Chermak-Delgado lattice is a quasi-antichain, J. Group Theory, 22(2019), no.3, 529-544.
  * [2] L. An, J. Brenna, H. Qu, E. Wilcox, Chermak-Delgado lattice extension theorems, Comm. Algebra , 43(2015), no.5, 2201-2213.
  * [3] B. Brewster, P. Hauck, E. Wilcox, Groups whose Chermak-Delgado lattice is a chain, J. Group Theory, 17(2014), no.2, 253-265.
  * [4] B. Brewster, P. Hauck, E. Wilcox, Quasi-antichain Chermak-Delgado lattice of finite groups, Arch. Math., 103(2014), no.4, 301-311.
  * [5] B. Brewster and E. Wilcox, Some groups with computable Chermak-Delgado lattices, Bull. Aust. Math. Soc., 86(2012), no.1, 29-40.
  * [6] E. Brush, J. Dietz, K. Johnson-Tesch and B. Power, On the Chermak-Delgado lattices of split metacyclic $p$-groups, Involve, 9(2016), no.5, 765-782.
  * [7] A. Chermak, A. Delgado, A measuring argument for finite groups, Proc. Amer. Math. Soc., 107(1989), no.4, 907–914.
  * [8] W. Cocke, Subnormality and the Chermak-Delgado lattice, J. Algebra Appl., 19(2020), no.8, 2050141, 7 pp.
  * [9] G. Glauberman, Centrally large subgroups of finite $p$-groups, J. Algebra, 300 (2006), 480-508.
  * [10] P. Hall, The classification of prime-power group, J. Reine Angew. Math., 182(1940), 130-141.
  * [11] I.M. Isaacs, Finite Group Theory, American Mathematical Society, 2008.
  * [12] R. McCulloch, Chermak-Delgado simple groups, Comm. Algebra, 45(2017), no.3, 983-991.
  * [13] R. McCulloch, Finite groups with a trivial Chermak-Delgado subgroup, J. Group Theory, 21(2018), no.3, 449-461.
  * [14] R. McCulloch and M. T$\check{a}$rn$\check{a}$uceanu, Two classes of finite groups whose Chermak-Delgado lattice is a chain of length zero, Comm. Algebra, 46 (2018), no.7, 3092-3096.
  * [15] A. Morresi Zuccari, V. Russo, and C.M. Scoppola, The Chermak-Delgado measure in finite $p$-groups, J. Algebra, 502(2018), 262-276.
  * [16] M. T$\check{a}$rn$\check{a}$uceanu, The Chermak-Delgado lattice of ZM-groups, Results Math., 72(2017), no.4, 1849-1855.
  * [17] M. T$\check{a}$rn$\check{a}$uceanu, A note on the Chermak-Delgado lattice of a finite group, Comm. Algebra, 46(2018), no.1, 201-204.
  * [18] M. T$\check{a}$rn$\check{a}$uceanu, Finite groups with a certain number of values of the Chermak-Delgado measure, J. Algebra Appl., 19 (2020), no.5, 2050088, 7 pp.