Copyright for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).

Forum for Information Retrieval Evaluation (FIRE)- 2023, Indian Statistical
Institute, Kolkata, India, $15^{th}-13^{th}$ December, 2023

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[1] [1]

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[1]Corresponding author. [1]These authors contributed equally.

# Generative AI for Software Metadata: Overview of the Information Retrieval
in Software Engineering Track at FIRE 2023

Srijoni Majumdar  Soumen Paul  Debjyoti Paul  Ayan Bandyopadhyay  Samiran
Chattopadhyay  Partha Pratim Das  Paul D Clough  Prasenjit Majumder  IIT
Kharagpur, West-Bengal, India  University of Leeds, UK  TCG CREST, West-
Bengal, India  TPXimpact London, UK  Sheffield University, Sheffield, UK  DA-
IICT Gandhinagar, Gujarat, India  Jadavpur University, West-Bengal, India
Indian Statistical Institute, Kolkata India

(2023)

###### Abstract

The Information Retrieval in Software Engineering (IRSE) track aims to develop
solutions for automated evaluation of code comments in a machine learning
framework based on human and large language model generated labels. In this
track, there is a binary classification task to classify comments as useful
and not useful. The dataset consists of 9048 code comments and surrounding
code snippet pairs extracted from open source github C based projects and an
additional dataset generated individually by teams using large language
models. Overall 56 experiments have been submitted by 17 teams from various
universities and software companies. The submissions have been evaluated
quantitatively using the F1-Score and qualitatively based on the type of
features developed, the supervised learning model used and their corresponding
hyper-parameters. The labels generated from large language models increase the
bias in the prediction model but lead to less over-fitted results.

###### keywords:

bert GPT-2 Stanford POS Tagging neural networks abstract syntax tree

##  1 Introduction

Assessing comment quality can help to de-clutter code bases and subsequently
improve code maintainability. Comments can significantly help to read and
comprehend code if they are consistent and informative.

The perception of quality in terms of the ’usefulness’ of the information
contained in comments is relative and hence is perceived differently based on
the context. Bosu et al. [1] attempted to assess code review comments (logged
in a separate tool) in the context of their utility in helping developers
write better code through a detailed survey at Microsoft. A similar quality
assessment model is important to analyse the type of source code comments that
can help for standard maintenance tasks but is largely missing. Majumdar et
al. [2] proposed a comment quality evaluation framework wherein comments were
assessed as ’useful’, ’partially useful’, and ’not useful’ based on whether
they increase the readability of the surrounding code snippets. The authors
analyse comments for concepts that aid in code comprehension and also the
redundancies or inconsistencies of these concepts with the related code
constructs in a machine learning framework for an overall assessment. The
concepts are derived through exploratory studies with developers across 7
companies and from a larger community using crowd-sourcing.

The first edition of the IRSE track of FIRE 2023, extends the work in [2] and
empirically investigates comment quality with a larger set of machine learning
solvers and features. The task is based on the quality evaluation of comments
into two clusters - ’useful’ and ’not useful’. A ’useful’ comment (refer Table
1) contains relevant concepts that are not evident from the surrounding code
design, and thus increases the comprehensibility of the code. The suitability
of analysing comment quality using various vector space representations of
code and comment pairs along with standard textual features and code comment
correlation links are evaluated.

Table 1: Useful and Not-Useful comments in context of code comprehension # |  Comment |  Code |  Label
---|---|---|---
1 |  /* uses png_calloc defined in pngriv.h*/ |  /* uses png_calloc defined in pngriv.h*/

    PNG_FUNCTION(png_const_structrp png_ptr)

    {  if (png_ptr == NULL || info_ptr == NULL)
          return;
       png_calloc(png_ptr); ...}

|  U
2 |  /* serial bus is locked before use */ |

    static int bus_reset ( . . . ) /* serial bus
    is locked before use*/
    { .. update_serial_bus_lock (bus * busR); }

|  NU
3 |  // integer variable |

    int Delete\_Vendor;  // integer variable

|  NU
U: Useful; NU: Not Useful

The 2023 IRSE track extends this challenge to understand the feasibility of
using silver standard quality labels generated from the Large Language Models
(LLMs) and understand how it augments the classification model in terms of
prediction. Developing the gold industry standard for analysing the usefulness
of comments that can be relevant for code comprehension in legacy systems can
be challenging and time-consuming. However, to scale the model and use it on
different languages, it is important to generate more data which we attempt to
do with the large language models. The performance of these modes in the
context of understating the relations between code and comment can provide an
approximation of the data quality generated and how it can be used to scale
the existing classification mode. This approach can also be further
generalised to any classification model based on software metadata.

##  2 Related Work

Software metadata is integral to code maintenance and subsequent
comprehension. A significant number of tools [3, 4, 5, 6, 7, 8] have been
proposed to aid in extracting knowledge from software metadata like runtime
traces or structural attributes of codes.

In terms of mining code comments and assessing the quality, authors [9, 10,
11] compare the similarity of words in code-comment pairs using the
Levenshtein distance and length of comments to filter out trivial and non-
informative comments. Rahman et al. [12] detect useful and non-useful code
review comments (logged-in review portals) based on attributes identified from
a survey conducted with developers of Microsoft [1]. Majumdar et al. [2, 13]
proposed a framework to evaluate comments based on concepts that are relevant
for code comprehension. They developed textual and code correlation features
using a knowledge graph for semantic interpretation of information contained
in comments. These approaches use semantic and structural features to design
features to set up a prediction problem for useful and not useful comments
that can be subsequently integrated into the process of decluttering
codebases.

With the advent of large language models [14], it is important to compare the
quality assessment of code comments by the standard models like GPT 3.5 or
llama with the human interpretation. The IRSE track at FIRE 2023 extends the
approach proposed in [2] to explore various vector space models [15] and
features for binary classification and evaluation of comments in the context
of their use in understanding the code. This track also compares the
performance of the prediction model with the inclusion of the GPT-generated
labels for the quality of code and comment snippets extracted from open-source
software.

##  3 IRSE Track Overview and Data Set

The following section outlines the task descriptions and the characteristics
of the dataset.

###  3.1 Task Description

Comment Classification: A binary classification task to classify source code
comments as Useful or Not Useful for a given comment and associated code pair
as input.

Input: A code comment with surrounding code snippet (written in C)

Output: A label (Useful or Not Useful) that characterises whether the comment
helps developers comprehend the associated code

Therefore in this classification task, the output is based on whether the
information contained in the comment is relevant _and_ would help to
comprehend the surrounding code, i.e., it is _useful_.

Useful: Comments have sufficient software development concept $\rightarrow$
Comment is Relevant, and these concepts are not mostly present in the
surrounding code $\rightarrow$ Comment is not Redundant, hence the comment is
Useful

Not Useful: Comments have sufficient software development concept
$\rightarrow$ Comment is Relevant, and these concepts are mostly present in
the surrounding code $\rightarrow$ Comment is Redundant, hence the comment is
Not Useful

It may also be the case that comments do not contain sufficient software
development concepts $\rightarrow$ Comment is Not Relevant, hence the comment
is Not Useful.

It is left to the participants to decide on the threshold value for how many
concepts retrieved make a comment relevant or how many matches with
surrounding code make a comment redundant.

The notion of relevant comments refers to those that developers perceive as
important in comprehending the associated or surrounding lines of code. These
concepts are related to the outline of the algorithm, data-structure
descriptions, mapping to user interface details, possible exceptions, version
details, etc. In the below examples, the comments highlight useful details
about the input data to the function, which is not evident from the associated
code itself.

Dataset: For the IRSE track, we use a set of 9048 comments (from Github) with
comment text, surrounding code snippets, and a label that specifies whether
the comment is useful or not. Sample data has been characterised in Table 1.

  * •

The development dataset contains 8048 rows of comment text, surrounding code
snippets, and labels (Useful and Not useful).

  * •

The test dataset contains 1,000 rows of comment text, surrounding code
snippets, and labels (Useful and Not useful).

Table 2: Characterizations of the Submissions: test Data Predictions Affiliation | Seed data | Seed data + LLM-generated data
---|---|---
Precision | Recall | F1Score | Precision | Recall | F1Score
DSTI, France | 0.8326 | 0.8626 | 0.8473 | 0.844 | 0.8682 | 0.8559
0.8948 | 0.8738 | 0.884 | 0.9 | 0.8707 | 0.885
0.8807 | 0.8822 | 0.8813 | 0.8871 | 0.8839 | 0.8854
SSN-1 (RAM) | 0.8 | 0.8 | 0.8 | 0.8021 | 0.81 | 0.73
0.72 | 0.71 | 0.74 | 0.7 | 0.73 | 0.74
SSN-2 (Aloy) | 0.788 | 0.7363 | 0.7613 | 0.89 | 0.8802 | 0.8846
0.7994 | 0.7994 | 0.7994 | 0.89 | 0.8795 | 0.8841
0.7993 | 0.9352 | 0.8619 | 0.839 | 0.9199 | 0.8776
0.7842 | 0.8453 | 0.8136 | 0.8154 | 0.8823 | 0.8475
0.7572 | 0.8637 | 0.807 | 0.7785 | 0.9003 | 0.835
| IIT (ISM)
---
Dhanbad
0.92 | 0.96 | 0.94 | 0.92 | 0.97 | 0.97
SSN-3 (Black) | 0.7916 | 0.8446 | 0.8172 | 0.7886 | 0.847 | 0.8167
0.763 | 0.8696 | 0.813 | 0.7655 | 0.8724 | 0.8154
0.705 | 0.9387 | 0.8052 | 0.6994 | 0.9041 | 0.7887
0.7292 | 0.856 | 0.7875 | 0.7374 | 0.8533 | 0.7911
| Microsoft-
---
American Express
0.7902 | 0.8016 | 0.7949 | 0.7908 | 0.8014 | 0.7952
DDU-1 | 0.895 | 0.891 | 0.893 | 0.890 | 0.894 | 0.892
DDU-2 | 0.875 | 0.872 | 0.874 | 0.870 | 0.875 | 0.880
IIT KGP-1 | 0.8283 | 0.804 | 0.8141 | 0.8322 | 0.8086 | 0.8185
SRM | 0.8283 | 0.804 | 0.8141 | 0.8178 | 0.7906 | 0.8013
IIT KGP-2 | 0.78 | 0.85 | 0.8 | 0.77 | 0.85 | 0.8
DA-IICT | 0.81 | 0.8 | 0.8 | 0.58 | 0.58 | 0.58
IIT Goa | 0.6087 | 0.6526 | 0.6321 | 0.6114 | 0.6598 | 0.6403
| TCS
---
0.778 | 0.753 | 0.74 | 0.645 | 0.6598 | 0.650
IIT KGP-3 | 0.631 | 0.645 | 0.639 | 0.6114 | 0.6598 | 0.631
Amazon | 0.659 | 0.672 | 0.666 | 0.656 | 0.635 | 0.645
Table 3: Characterizations of the LLM Generated datasets Team name | Total entry | Useful entry | Not useful entry
---|---|---|---
DSTI, France | 421 | 412 | 9
SSN-1 (RAM) | 1238 | 740 | 497
SSN-2 (Alloy) | 1510 | 24 | 1486
IIT (ISM) Dhanbad | 199 | 182 | 17
SSN 3 (Black) | 738 | 80 | 658
Microsoft - American Express | 233 | 92 | 141
DDU-1 | 8588 | 4649 | 3939
DDU-2 | 332 | 311 | 21
IIT KGP-1 | 334 | 309 | 25
SRM | 217 | 196 | 21
IITKGP-2 | 263 | 130 | 133
DA-IICT | 150 | 65 | 85
IIT-Goa | 543 | 460 | 83
TCS | 282 | 61 | 221
IITKGP-3 | 570 | 450 | 120
IITKGP-3 | 412 | 345 | 67

##  4 Participation and Evaluation

IRSE 2023 received a total of 56 experiments from 17 teams for the two tasks.
As this track is related to software maintenance, we received participation
from companies like Microsoft, Amazon, Amercian Express, Bosch Research along
with several research labs of educational institutes.

The various teams with the details of their submissions are characterised in
Table 2.

Evaluation Procedure: Candidates submited the prediction metrics
(precision,recall, F1-Score) fo the classification model with the Gold labels
dataset (referred to as the Seed Dataset) and combined dataset (Seed + LLM
generated labels - Silver lables dataset). The difference in the F1 score was
evaluated by us.

Features: Apart from evaluating the prediction metrics, we analysed the types
of features the teams have used to devise the machine learning pipeline. The
teams have performed routine pre-processing and have retained the significant
words or letters only for both the code and comment pairs. Further, some of
the teams have also used morphological features of a comment like a length,
significant words ratio, parts of speech characteristics, or occurrence of
words from an enumerated set as textual features. To correlate code and
comment and detect redundancies, the teams mostly used grep-like string match
to find similar words.

Vector Space Representations: Code and comments belong- to different semantic
granularity which is unified by a vector space representation. The
participants have used various pre-trained embeddings to generate vectors for
the words like those based on one hot encoding, tf-idf based, word2vec or
context aware like ELMo and BERT. Each of the employed embedding models are
trained or finetuned using software development corpora.

Results: The participants are able to achieve a slight increase (in the range
of 2%-4%) in the test prediction metrics and in many cases the performance
decrease. However, the increase in bias due to the incorporation of silver
standard data reduces the over-fitting of the models.

##  5 Conclusions

The IRSE 2023 track empirically investigates the feasibility of augmenting
existing classification models using datasets with labels generated from
LLM’s. A total of 17 teams participated and submitted 56 experiments that used
various types of machine learning models, embedding spaces, features and
different LLMs to generate data. The LLM-generated labels reduce the
overfitting of the overall classification model and also improve the F1 score
when the combined data from all participants were used to augment the existing
data with gold standard labels from industry practitioners.

## References

  * Bosu et al. [2015] A. Bosu, M. Greiler, C. Bird,  Characteristics of useful code reviews: An empirical study at microsoft,  Working Conference on Mining Software Repositories, IEEE, 2015, pp. 146–156.
  * Majumdar et al. [2022] S. Majumdar, A. Bansal, P. P. Das, P. D. Clough, K. Datta, S. K. Ghosh,  Automated evaluation of comments to aid software maintenance,  Journal of Software: Evolution and Process 34 (2022) e2463.
  * Majumdar et al. [2019] S. Majumdar, S. Papdeja, P. P. Das, S. K. Ghosh,  Smartkt: a search framework to assist program comprehension using smart knowledge transfer,  in: 2019 IEEE 19th International Conference on Software Quality, Reliability and Security (QRS), IEEE, 2019, pp. 97–108.
  * Chatterjee et al. [2015] N. Chatterjee, S. Majumdar, S. R. Sahoo, P. P. Das,  Debugging multi-threaded applications using pin-augmented gdb (pgdb),  in: International conference on software engineering research and practice (SERP). Springer, 2015, pp. 109–115.
  * Majumdar et al. [2016] S. Majumdar, N. Chatterjee, S. R. Sahoo, P. P. Das,  D-cube: tool for dynamic design discovery from multi-threaded applications using pin,  in: 2016 IEEE International Conference on Software Quality, Reliability and Security (QRS), IEEE, 2016, pp. 25–32.
  * Majumdar et al. [2021a] S. Majumdar, N. Chatterjee, P. P. Das, A. Chakrabarti,  A mathematical framework for design discovery from multi-threaded applications using neural sequence solvers,  Innovations in Systems and Software Engineering 17 (2021a) 289–307.
  * Majumdar et al. [2021b] S. Majumdar, N. Chatterjee, P. Pratim Das, A. Chakrabarti,  Dcube_ nn d cube nn: Tool for dynamic design discovery from multi-threaded applications using neural sequence models,  Advanced Computing and Systems for Security: Volume 14 (2021b) 75–92.
  * O’brien [2003] M. P. O’brien, Software comprehension–a review and research direction, Technical Report Technical Report, Department of Computer Science & Information Systems University of Limerick, Ireland, 2003.
  * Steidl et al. [2013] D. Steidl, B. Hummel, E. Juergens,  Quality analysis of source code comments,  International Conference on Program Comprehension (ICPC), IEEE, 2013, pp. 83–92.
  * Majumdar et al. [2022a] S. Majumdar, A. Bandyopadhyay, P. P. Das, P. Clough, S. Chattopadhyay, P. Majumder,  Can we predict useful comments in source codes?-analysis of findings from information retrieval in software engineering track@ fire 2022,  in: Proceedings of the 14th Annual Meeting of the Forum for Information Retrieval Evaluation, 2022a, pp. 15–17.
  * Majumdar et al. [2022b] S. Majumdar, A. Bandyopadhyay, S. Chattopadhyay, P. P. Das, P. D. Clough, P. Majumder,  Overview of the irse track at fire 2022: Information retrieval in software engineering,  in: Forum for Information Retrieval Evaluation, ACM, 2022b.
  * Rahman et al. [2017] M. M. Rahman, C. K. Roy, R. G. Kula,  Predicting usefulness of code review comments using textual features and developer experience,  International Conference on Mining Software Repositories (MSR), IEEE, 2017, pp. 215–226.
  * Majumdar et al. [2020] S. Majumdar, S. Papdeja, P. P. Das, S. K. Ghosh,  Comment-mine - a semantic search approach to program comprehension from code comments,  in: Advanced Computing and Systems for Security, Springer, 2020, pp. 29–42.
  * Brown et al. [2020] T. Brown, B. Mann, N. Ryder, M. Subbiah, J. D. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, et al.,  Language models are few-shot learners,  Advances in neural information processing systems 33 (2020) 1877–1901.
  * Majumdar et al. [2022] S. Majumdar, A. Varshney, P. P. Das, P. D. Clough, S. Chattopadhyay,  An effective low-dimensional software code representation using bert and elmo,  in: 2022 IEEE 22nd International Conference on Software Quality, Reliability and Security (QRS), IEEE, 2022, pp. 763–774.


00footnotetext: 2010 AMS Subject Classification: Primary 20K10, Secondary
20K12. Key words and phrases: Abelian groups, characteristic subgroups, fully
invariant subgroups, strongly invariant subgroups.

# On Abelian Groups Having Isomorphic
Proper Strongly Invariant Subgroups

Andrey R. Chekhlov
Faculty of Mathematics and Mechanics, Section of Algebra,
Tomsk State University, Tomsk 634050, Russia
e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
and
Peter V. Danchev
Institute of Mathematics and Informatics, Section of Algebra,
Bulgarian Academy of Sciences, Sofia 1113, Bulgaria
e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider two variants of those Abelian groups with all proper strongly
invariant subgroups isomorphic and give an in-depth study of their basic and
specific properties in either parallel or contrast to the Abelian groups with
all proper fully invariant (respectively, characteristic) subgroups
isomorphic, which are studied in details by the current authors in Commun.
Algebra (2015) and in J. Commut. Algebra (2023). In addition, we also explore
those Abelian groups having at least one proper strongly invariant subgroup
isomorphic to the whole group.

##  1 Introduction and Definitions

Throughout the present paper, let all groups into consideration be additively
written and Abelian. Our notations and terminology from group theory are
mainly standard and follow those from [9], [10] and [14], respectively.
Another useful sources on the explored subject are [4, 5, 6, 7] as well. For
instance, if $p$ is a prime integer and $G$ is an arbitrary group,
$p^{n}G=\\{p^{n}g~{}|~{}g\in G\\}$ denotes the $p^{n}$-th power subgroup of
$G$ consisting of all elements of $p$-height greater than or equal to
$n\in\mathbb{N}$, $G[p^{n}]=\\{g\in G~{}|~{}p^{n}g=0,n\in\mathbb{N}\\}$
denotes the $p^{n}$-socle of $G$, and $G_{p}=\cup_{n<\omega}G[p^{n}]$ denotes
the $p$-component of the torsion part $tG=\oplus_{p}G_{p}$ of $G$.

On the other hand, if $G$ is a torsion-free group and $a\in G$, then let
$\chi_{G}(a)$ denote the characteristic and let $\tau_{G}(a)$ denote the type
of $a$, respectively. Specifically, the class of equivalence in the set of all
characteristics is just called type and we write $\tau$. If
$\chi_{G}(a)\in\tau$, then we write $\tau_{G}(a)=\tau$, and so
$\tau(G)=\\{\tau_{G}(a)~{}|~{}0\neq a\in G\\}$ is the set of types of all non-
zero elements of $G$. The set $G(\tau)=\\{g\in
G~{}|~{}\tau(g)\geqslant\tau\\}$ forms a pure fully invariant subgroup of the
torsion-free group $G$. Recall that a torsion-free group $G$ is called
homogeneous if all its non-zero elements have the same type.

Concerning ring theory, suppose that all rings which we consider are
associative with identity element. For any ring $R$, the letter $R^{+}$ will
denote its additive group. To simplify the notation and to avoid a risk of
confusion, we shall write ${\rm E}(G)$ for the endomorphism ring of $G$ and
$\mathop{\rm End}\nolimits(G)={\rm E}(G)^{+}$ for the endomorphism group of
$G$.

As usual, a subgroup $F$ of a group $G$ is called fully invariant (hereafter
abbreviated as a fi-subgroup for simpleness) if $\phi(F)\subseteq F$ for any
$\phi\in{\rm E}(G)$, while if $\phi$ is an invertible endomorphism (= an
automorphism), then $F$ is called a characteristic subgroup. Likewise,
imitating [1], we shall say that a subgroup $S$ of $G$ is strongly invariant,
provided that $\psi(S)\subseteq S$ for any homomorphism $\psi:S\rightarrow G$,
and in what follows we shall abbreviate it for short by a si-subgroup. It is
well-known that the following relations are fulfilled:

strongly invariant $\Rightarrow$ fully invariant $\Rightarrow$ characteristic.

Classical examples of important fully invariant subgroups of an arbitrary
group $G$ are the defined above subgroups $p^{n}G$ and $G[p^{n}]$ for any
natural $n$ as well as $tG$ and the maximal divisible subgroup $dG$ of $G$;
actually $dG$ is a fully invariant direct summand of $G$ (see, for instance,
[9]). To avoid any confusion and misunderstanding, we shall say that a group
$G$ has only trivial fully invariant subgroups if $\\{0\\}$ and $G$ are the
only ones. Same appears for the characteristic and strongly invariant
subgroups, respectively.

Let us notice that the si-subgroups were intensively studied in [1, 3],
respectively, as well as in some other relevant articles cited therewith.

Note also that, for all subgroups $F\leq G$, the subgroup

$\mathrm{Hom}(F,G)F=\sum_{\varphi\in\mathrm{Hom}(F,G)}\varphi(F)$

is the minimal si-subgroup of $G$ containing $F$. In particular, if $F$ is a
si-subgroup, then $F=\mathrm{Hom}(F,G)F$. Likewise, we also set

$S_{A}(B)=\sum_{f\in\mathrm{Hom}(A,B)}\mathop{\rm Im}\nolimits f$

to be the $A$-_socle_ of $B$.

The following key notions, necessary for our successful presentation, were
stated in [4].

Definition 1. A non-zero group $G$ is said to be an IFI-group if either it has
only trivial fully invariant subgroups, or all its non-trivial fully invariant
subgroups are isomorphic otherwise.

Definition 2. A non-zero group $G$ is said to be an IC-group if either it has
only trivial characteristic subgroups, or all its non-trivial characteristic
subgroups are isomorphic otherwise.

Note that Definition 2 implies Definition 1. In other words, any IC-group is
an IFI-group; in fact every fully invariant subgroup is characteristic.

Definition 3. A non-zero group $G$ is said to be a strongly IFI-group if
either it has only trivial fully invariant subgroups, or all its non-zero
fully invariant subgroups are isomorphic otherwise.

Definition 4. A non-zero group $G$ is said to be a strongly IC-group if either
it has only trivial characteristic subgroups, or all its non-zero
characteristic subgroups are isomorphic otherwise.

Notice that Definition 4 implies Definition 3.

On another vein, Definition 4 obviously yields Definition 2, but the converse
is false. In fact, in [14] was constructed a single non-trivial characteristic
subgroup of a $2$-group that are pairwise non-isomorphic, thus giving an
example of an IC-group which is surely not a strongly IC-group; in other
words, Definition 4 properly implies Definition 2.

We now arrive at our basic tools as follows:

Definition 5. A non-zero group $G$ is called an ISI-group if either it has
only trivial strongly invariant subgroups (namely, $\\{0\\}$ or $G$), or all
its non-trivial strongly invariant subgroups are isomorphic otherwise.

Definition 6. A non-zero group $G$ is called a strongly ISI-group if either it
has only trivial strongly invariant subgroups, or all its non-zero strongly
invariant subgroups are isomorphic otherwise.

It is clear that any strongly ISI-group is either a $p$-group for some prime
$p$, or is a homogeneous torsion-free group.

It is apparent that the next relationships are valid:

IC-groups $\Rightarrow$ IFI-groups $\Rightarrow$ ISI-groups.

Our objective in the current article is to explore some fundamental and exotic
properties of the defined above classes of groups, especially the ISI-groups
and the strongly ISI-groups. In addition, we shall investigate even something
more, namely the existence of a non-trivial strongly invariant subgroup of a
given group, which subgroup is isomorphic to the whole group, calling these
groups weakly ISI-groups. Our motivation to do that is to exhibit and compare
certain similarities and discrepancies of these group classes.

The major results established by us are formulated and proved in the next
section. Concretely, our work is organized thus: In the next section, we
provide our basic statements and their proofs by breaking the results into
three parts in conjunction with the definitions stated above and pertained to
the three types of the ISI property, namely to the so-termed (strongly,
weakly) ISI-groups. In order to prove our results, we shall use some specific
ideas and instruments to materialize them (see Theorems 2.11, 2.13 as well as
the corresponding lemmas, propositions and examples). We end off our
presentation with a series of six questions which, hopefully, will motivate a
further intensive study of the explored subject (see Problems 1-6).

##  2 Main Theorems and Examples

For completeness of the exposition and for the reader’s convenience, we first
and foremost will give a brief retrospection of the most principal results
achieved in [4] and [7], respectively, concerning IFI-groups and strongly IFI-
groups as well as IC-groups and strongly IC-groups, respectively.

As usual, the symbol $\oplus_{m}G=G^{({m})}$ will denote the external direct
sum of $m$ copies of the group $G$, where $m$ is some ordinal (finite or
infinite).

We begin with a retrospection of some of the results obtained in [4].

###### Theorem 2.1

Let $G$ be a $p$-group and let $m\geqslant 2$ be an ordinal. Then $G^{({m})}$
is an IFI-group if, and only if, $G$ is an IC-group.

###### Proposition 2.2

Let $G$ be a torsion-free group. Then $G$ is an IFI-group if, and only if, $G$
is a strongly IFI-group.

###### Lemma 2.3

(a) A fully invariant subgroup of an IFI-group is an IFI-group.

(b) A fully invariant subgroup of a strongly IFI-group is a strongly IFI-
group.

###### Proposition 2.4

A non-zero IFI-group is either divisible or reduced.

###### Theorem 2.5

The following two points hold:

(i) A non-zero group $G$ is an IFI-group if, and only if, one of the following
holds:

$\bullet$ For some prime $p$ either $pG=\\{0\\}$, or $p^{2}G=\\{0\\}$ with
$\mathrm{rank}(G)=\mathrm{rank}(pG)$.

$\bullet$ $G$ is a homogeneous torsion-free IFI-group of idempotent type.

(ii) A non-zero torsion group $G$ is a strongly IFI-group if, and only if, it
is an elementary $p$-group for some prime $p$.

###### Proposition 2.6

Every homogeneous fully transitive torsion-free group of idempotent type is an
IFI-group.

###### Corollary 2.7

A direct summand of a fully transitive torsion-free IFI-group is again a fully
transitive IFI-group.

We now continue with some of the most important corresponding assertions from
[7].

###### Proposition 2.8

Every torsion IFI-group is an IC-group.

###### Theorem 2.9

There exists an IFI-group which is not an IC-group.

###  2.1 ISI-groups

We start our work here with some elementary but useful observations:

###### Proposition 2.10

Let $G$ be a non si-simple group. Then $G$ is an ISI-group if, and only if,
$G$ has a single non-trivial si-subgroup $H$.

Proof. ”$\Rightarrow$”. Assume that both $H$ and $F$ are non-trivial si-
subgroups. Thus, $H\cong F$, so $F\leq\mathrm{Hom}(H,G)H$ and hence $F\leq H$.
Analogously, we derive that $H\leq F$, i.e., $H=F$, as required.

”$\Leftarrow$”. It is evident, so we omit the details.

The next Theorem is quite similar to [4, Theorem 2.5], but however is somewhat
its reminiscent.

###### Theorem 2.11

A non-zero torsion group is an ISI-group if, and only if, for some prime $p$,
either $pG=\\{0\\}$ or $p^{2}G=\\{0\\}$.

Proof. Suppose first that $G$ is torsion, that is, $G=tG$. If $G=G[p]$, the
assertion follows automatically. So, assume next that $G\neq G[p]$. We now
assert that $G=G[p^{2}]$. If $G\neq G[p^{2}]$, then $G[p]\cong G[p^{2}]$ which
is an absurd, so that the claim is sustained.

Reciprocally, if $G$ is an elementary $p$-group, then it contains only trivial
si-subgroups, and thus we are done. So, let $G$ be $p^{2}$-pounded. It is well
known in this case that the only proper si-subgroups of $G$ is $G[p]$, as
needed.

In some cases the subgroup $H$ from Proposition 2.10 is si-simple (which is
the case not only in Theorem 2.11).

###### Proposition 2.12

If $G=A\oplus B$, where $A\neq\\{0\\}$ is fully invariant in $G$, then $G$ is
an ISI-group if, and only if, $A$ and $B$ are such si-simple groups that
$\mathrm{Hom}(B,A)\neq\\{0\\}$.

Proof. ”$\Rightarrow$”. If we assume in a way of contradiction that $A$ is not
a si-simple group, then it will contain an si-subgroup, say $\\{0\\}\neq F\neq
A$. But then both $A$ and $F$ are simultaneously si-subgroups in $G$, so that,
by assumption, we can deduce that $F\cong A$. In fact, if
$S=\mathrm{Hom}(F,B)F$, then $A\oplus S$ is a si-subgroup in $G$, whence
$A\cong A\oplus S$, but thus contradicting $\mathrm{Hom}(A,B)=\\{0\\}$. So,
with no harm of generality, we may assume that $A\leq F$, i.e., $F=A$, which
is the desired contradiction.

Also, if $\mathrm{Hom}(B,A)=\\{0\\}$, then both $A$ and $B$ are non-isomorphic
si-subgroups. If, however, $\\{0\\}\neq C\neq B$ is a si-subgroup, then
$A\oplus C\cong A$ and, therefore, $\mathrm{Hom}(A,B)\neq\\{0\\}$, again a
contradiction, whence $B$ is si-simple, as wanted.

”$\Leftarrow$”. Let $\\{0\\}\neq H\leq G$ be a si-subgroup and let
$\pi:G\rightarrow B$ be the corresponding projection. If $H\leq A$, then $H$
is just a si-subgroup in $A$, and so $H\cong A$. Assuming now that $H\nleq A$,
then the relations $\\{0\\}\nleq\pi(H)\leq B$ hold. So,

$B\leq\mathrm{Hom}(\pi(H),B)\pi(H)\leq\mathrm{Hom}(\pi(H),G)\pi(H)\leq H$

and, consequently, $H=B\oplus(H\cap A)$, where $H\cap A$ is a si-subgroup in
$A$. Finally, $H\cap A=A$ and thus $H=G$, as pursued.

It follows from Proposition 2.12 that the sufficiency in the following two
statements are directly true.

###### Theorem 2.13

The mixed group $G$ is an ISI-group if, and only if, $G=T\oplus R$, where $T$
is an elementary $p$-group for some $p$ and $R$ is such a si-simple torsion-
free group that $pR\neq R$.

Proof. ”$\Rightarrow$”. It is clear that the torsion part $tG$ of such a group
$G$ is an elementary $p$-group, so one can decompose $G=T\oplus R$, where
$T=tG$ and $R$ is torsion-free. If $F\neq\\{0\\}$ is a si-subgroup in $R$ and
$F\neq R$, then $T$ and $T\oplus F$ are apparently non-isomorphic si-subgroups
of $G$, so that $R$ is a si-simple group, as claimed.

###### Proposition 2.14

The non-reduced group $G$ is an ISI-group if, and only if, $G=D\oplus R$,
where $D$ is a torsion-free divisible group and $R$ is a si-simple torsion-
free group.

Proof. ”$\Rightarrow$”. Clearly, the divisible part of $G$ is torsion-free (by
following the same arguments as in Theorem 2.13) and, moreover, we may apply
the same Theorem 2.13 to get that the group $R$ is a si-simple, as asserted.

We shall say that a torsion-free group $G$ is strongly irreducible (or,
shortly, s-irreducible) if $G$ does not have proper pure si-subgroups.

It is very clear that if $G$ is a torsion-free group and $nG=G$ for some
$n\in\mathbb{N}$, then $nH=H$ for each fi-subgroup $H$ of $G$. This follows
immediately from the fact that $n^{-1}\cdot 1_{G}\in\mathrm{E}(G)$.

The next two examples are crucial for our considerations.

###### Example 2.15

There exists a non si-simple homogeneous torsion-free group.

Proof. Let $A$ be a module on the ring $\widehat{\mathbb{Z}}_{p}$ consisting
of $p$-adic integers such that, as a group, $A$ is reduced torsion-free, and
let $B$ be a separable homogeneous torsion-free group of type
$\tau=\tau(\widehat{\mathbb{Z}}^{+}_{p})$. Hence, $G=A\oplus B$ is homogeneous
and also it is readily checked that $G$ is an ISI-group which is not si-
simple, because $A$ is a si-direct summand in $G$.

###### Example 2.16

There exists an s-irreducible group that is not an irreducible group.

Proof. Let $G$ be the group from [10, § 88, Exersize 6]. Then,
$\mathrm{E}(G)=\mathbb{Z}$ and the rank of $G$ can be chosen to be a free
natural number $n\geq 2$. So, every subgroup of $G$ is fully invariant and,
consequently, $G$ is not irreducible. Likewise, each subgroup $A$ in $G$ of
rank $1\leq rank(A)\leq n-1$ is a free group, so that $\mathrm{Hom}(A,G)A=G$.
But, if the subgroup $B$ is of rank $n$, then one inspects that $B$ is
essential in $G$, whence $F=\mathrm{Hom}(A,G)A=G$ is also essential in $G$ and
thus $F_{*}=G$. Therefore, $G$ is s-irreducible. Note also that since all pure
subgroups in $G$ of rank $\leq n-1$ are free, they are not si-subgroups, thus
answering in the negative question (1), posed in [1].

The following technicalities are worthwhile for further applications.

###### Proposition 2.17

If $G$ is a torsion-free ISI-group and $\\{0\\}\neq H\lneqq G$ is a si-
subgroup, then $H$ is $q$-pure in $G$ for all prime numbers $q$ except, maybe,
one prime number $p$. Moreover, if $H$ is not $p$-pure in $G$ for some prime
$p$, then $pG\lneqq H$.

Proof. Assume that $pg\in H$ for some prime $p$, where $g\in G\setminus H$.
Letting

$F=\mathrm{Hom}(\langle H,g\rangle,G)\langle H,g\rangle,$

then $F$ is a si-subgroup in $G$, so either $F\cong H$ or $F=G$. Besides, the
condition $F\cong H$ yields $H=F$ that contradicts the fact that $g\in
G\setminus H$. So, it follows that $F=G$.

Furthermore, each element $y\in G$ can be written in the form

$y=f_{1}(x_{1}+l_{1}g)+\dots+f_{n}(x_{n}+l_{n}g)$

for some $f_{i}\in\mathrm{Hom}(\langle H,g\rangle,G)$, $x_{i}\in H$ and
$l_{i}\in\mathrm{Z}$; $i=1,\dots,n$. That is why,

$py=f_{1}(px_{1}+pl_{1}g)+\dots+f_{n}(px_{n}+pl_{n}g)\in H$

since $px_{i}+pl_{i}g\in H$. So, $pG\leq H$, where $pG\neq H$ since $pG\cong
G$ and $H\neq G$. If, however, $H$ is not $q$-pure for some prime $q\neq p$,
then similarly $qG\leq H$, which implies that $H=G$, so $H$ is indeed $q$-pure
for all primes $q\neq p$.

###### Remark 2.18

Let us note that Proposition 2.14 and Example 2.15 manifestly give valuable
examples of proper pure si-subgroups in torsion-free ISI-groups. Also, note
that the si-subgroup $H$ in Proposition 2.17 has the following property:

$\mathrm{Hom}(A,G)A=H,$

provided

$\\{0\\}\neq\mathrm{Hom}(A,G)A\neq G$

for every group $A$.

The next assertion is rather curious by comparing with the inheritance of the
direct summand property by the IFI and IS groups, respectively.

###### Lemma 2.19

A direct summand of an ISI-group $G$ is again an ISI-group.

Proof. Write $G=A\oplus B$ and given $S\leq A$ is a si-subgroup in $A$. If we
set $F=\mathrm{Hom}(S,B)S$, then $S\oplus F$ is obviously a si-subgroup in
$G$. However, if $S\oplus F\neq G$, then according to Proposition 2.10 we
conclude that $S\oplus F$ is the only non-trivial si-subgroup. Hence, it
follows that $A$ is an ISI-group, as formulated.

We also notice that it follows immediately from Theorem 2.13 and Lemma 2.19
that the following is true:

###### Proposition 2.20

The non-zero divisible group $D$ is an ISI-group if, and only if, $D$ is
torsion-free.

We are now ready to give our two desired examples.

###### Example 2.21

Pure si-subgroups of an ISI-group $G$ are not necessarily direct summands of
$G$.

Proof. Let $A$ and $B$ be both torsion-free groups of rank $1$ with
$\tau(A)<\tau(B)=\tau$, and let

$G=\langle A\oplus B,p^{-1}(a+b)\,|\,a\in A\setminus pA,b\in B\setminus
pB\rangle.$

Then, $B$ is pure in $G$ and $B=G(\tau)$, so $B$ is a si-subgroup in $G$.
Suppose that $\pi:A\oplus B\rightarrow A$ is a projection. If $H\neq B$ is a
si-subgroup of $G$, then one deduces that $H\nleqslant B$ since $B$ is si-
simple, and hence $\pi(pH)\neq\\{0\\}$. Thus, it follows that $A\leq H$ and so
$H=G$, as expected.

Notice that an example of si-subgroups in non-homogeneous torsion-free groups
is given in the previous Example 2.21, and an example of pure si-subgroups in
homogeneous torsion-free groups is given in Example 2.15.

###### Example 2.22

There is a homogeneous torsion-free group $A$ of rank $r\geq 2$ with a si-
subgroup $H$ as exhibited in Proposition 2.17.

Proof. Let $A$ be the group as constructed in [10, Example 5 from §88], that
is,

$A=\langle a_{1},\dots,a_{r},x_{1},x_{2},\dots\rangle,$

where $x_{n}=p^{-n}(a_{1}+\pi_{2n}a_{2}+\dots+\pi_{rn}a_{r})$,
$\pi_{in}=s_{i0}+s_{i1}p+\dots+s_{i,n-1}p^{n-1}$ is the $n-1$ partial sum of
the $p$-adic reversible number $\pi_{i}$ ($i=2,\dots,r$), $\pi_{i}$ are
algebraically independent on $\mathbb{Q}$ and $\pi_{1}=1$. It is principally
known that $\mathrm{E}(A)=\mathbb{Z}$, the subgroup $\langle
a_{2},\dots,a_{r}\rangle$ is pure in $A$ and all of the elements from
$A\setminus A_{0}$ are of the form $kx_{n}+k_{2}a_{2}+\dots+k_{r}a_{r}$, where
$k,k_{2},\dots,k_{r}\in\mathbb{Z}$, $p\nmid k$ and $n$ is some natural number.
Letting

$H=\langle a_{1},a_{2}\dots,a_{r},px_{1},px_{2},\dots\rangle,$

it is clear then that $pA\leq H$ and $H\neq A$.

We now manage to show that $\mathrm{Hom}(H,A)=\mathrm{Z}$. To that goal, we
shall use some techniques from [10, Example 5 of §88]. And so, choose
$0\neq\eta\in\mathrm{Hom}(H,A)$ and define the homomorphism $\eta$ thus:

$\eta a_{i}=\sum_{j=1}^{r}t_{ij}a_{j},~{}t_{ij}\in\mathbb{Z}.$

Notice that, if required, multiply $\eta$ on some $m\in\mathbb{N}$ so that the
new map will send $A_{0}=\langle a_{1},\dots,a_{r}\rangle$ into itself.
Therefore,

$\eta(px_{n})=p^{1-n}\sum_{i}\pi_{in}\eta
a_{i}=p^{1-n}\sum_{j}\Bigl{(}\sum_{i}\pi_{in}t_{ij}\Bigr{)}a_{j}=k_{n}px_{n}+l_{2n}a_{2}+\dots+l_{rn}a_{r}$

for some $k_{n},l_{2n},\dots,l_{rn}\in\mathbb{Z}$. We also have

$\sum_{i=1}^{r}\pi_{in}t_{i1}=k_{n}\ \mathrm{and}\
p^{1-n}\sum_{j=2}^{r}\Bigr{[}\sum_{i=1}^{r}\pi_{in}t_{ij}-k_{n}\pi_{jn}\Bigr{]}a_{j}\in\langle
a_{2},\dots,a_{r}\rangle.$

Furthermore, one readily can verify that the coefficients in the square
brackets are divided by $p^{n-1}$. Also, under taking $n\rightarrow\infty$,
for each index $j=2,\dots,r$ we obtain that
$\sum_{i=1}^{r}\pi_{i}t_{ij}-\kappa\pi_{j}=0$, where
$\sum_{i=1}^{r}\pi_{i}t_{i1}=\kappa$.

Now, in view of the algebraic independence of the $\pi_{i}$’s, it follows that
$t_{jj}=t_{11}$ and $t_{ij}=0$ assuming $i\neq j$. Thereby, the homomorphism
$\eta$ acts as multiplying on the integer $t_{11}$. Since
$\chi(a_{i})=(0,0,\dots)$, there is no problem to take $m>1$. So, we infer
that $\mathrm{Hom}(H,A)=\mathbb{Z}$, as asked for, and hence $H$ is a si-
subgroup in $A$, as wanted.

The following technicality is pretty simple but useful.

###### Lemma 2.23

(1) If $G$ is a torsion-free group and $H\leq G$ is a si-subgroup, then
$H_{*}$ also is a si-subgroup in $G$.

(2) A direct summand of s-irreducible (si-simple) group $G$ is again an
s-irreducible (si-simple).

Proof. (1) Letting $f\in\mathrm{Hom}(H_{*},G)$ and $x\in H_{*}$, we have
$nx\in H$ for some $n\in\mathbb{N}$, so that $f(nx)=nf(x)\in H$ whence
$f(x)\in H_{*}$.

(2) Write $G=A\oplus B$ and assume that $H\leq A$ is a si-subgroup in $A$ and
put $F=\mathrm{Hom}(H,B)H$. Thus, $H\oplus F$ is a si-subgroup in $G$. If,
however, $H\neq A$, then one has that $H\oplus F\neq G$ that contradicts the
condition that the group $G$ is si-simple.

Similarly, if $H_{*}\neq A$, then we will have that $H_{*}\oplus F_{*}\neq G$
which contradicts the condition that the group $G$ is s-irreducible.

The next two statements are worthy of noticing.

###### Proposition 2.24

Let $G$ be a torsion-free s-irreducible group which is not a si-simple ISI-
group. Then its non-trivial si-subgroup $H$ is also s-irreducible.

Proof. Owing to Proposition 2.17, we derive that $pG<H$ for some prime $p$.
Let $S\neq\\{0\\}$ be a pure si-subgroup in $H$ and $F=\mathrm{Hom}(S,G)S$.
Then, one sees that either $F=H$ or $F=G$. If, for a moment, $F=H$, then

$H=\mathrm{Hom}(S,G)S=\mathrm{Hom}(S,H)S=S.$

Assume now that $F=G$. Therefore,

$pG=\sum_{f\in\mathrm{Hom}(S,G)}pf(S)=\sum_{f\in\mathrm{Hom}(S,H)}pf(S)$

since all $pf(S)\leq H$. Further, $pG\leq S$ as
$\sum_{f\in\mathrm{Hom}(S,H)}f(S)=S$. In particular, $pH\leq S$ and so $S=H$
in view of purity of $S$ in $H$, as promised.

The next assertion is closely relevant to Example 2.21.

###### Proposition 2.25

(1) The pure si-subgroup of a direct sum $G$ of s-irreducible torsion-free
groups coincides with some fully invariant direct summand of $G$.

(2) The direct sum of s-irreducible torsion-free groups $A_{i}$, where $i\in
I$, $|I|\geq 2$, is an s-irreducible group if, and only if,
$\mathrm{Hom}(A_{i},A_{j})\neq\\{0\\}$ for all indices $i,j\in I$.

(3) The direct sum $G$ of si-simple groups $A_{i}$, where $i\in I$, $|I|\geq
2$, is a si-simple group if, and only if,
$\mathrm{Hom}(A_{i},A_{j})\neq\\{0\\}$ for all indexes $i,j\in I$.

Proof. (1) Let us write $G=\bigoplus_{i\in I}A_{i}$, where all $A_{i}$ are
s-irreducible groups and $H\leq G$ is a si-subgroup. Then, $H=\bigoplus_{i\in
I}(H\cap A_{i})$. However, since $H_{*}=\bigoplus_{i\in I}(H\cap A_{i})_{*}$,
we have that $H_{*}=\bigoplus_{j\in J}A_{j}$, where $J\subseteq I=\\{j\in
I\,|\,H\cap A_{j}\neq\\{0\\}\\}$, as required.

(2) ”$\Rightarrow$”. Assume that $\mathrm{Hom}(A_{i},A_{j})=\\{0\\}$ for some
$i\neq j$. Setting

$S=\\{s\in I\,|\,\mathrm{Hom}(A_{i},A_{s})\neq\\{0\\}\\},$

we observe that $S\neq\emptyset$, because $i\in S$ and $S\neq I$ since $j\in
I\setminus S$, so $\bigoplus_{s\in S}A_{s}$ is a proper pure si-subgroup
(i.e., $\neq\\{0\\},G$) that contradicts the s-irreducibility of $G$.

”$\Leftarrow$”. Given $H$ is a pure si-subgroup in $G$. We thus may write
$H=\bigoplus_{i\in I}(H\cap A_{i})$ and $H\cap A_{i}=A_{i}$ with $H\cap
A_{i}\neq\\{0\\}$ which holds in view of s-irreducibility of $A_{i}$ for each
index $i$. So, one follows that $H=G$ as
$\mathrm{Hom}(A_{i},A_{j})\neq\\{0\\}$ for all $i,j\in I$.

(3) The proof is similar to that in (2).

Furthermore, in virtue of Proposition 2.12, in the next statement the
fulfillment of the inequalities $\mathrm{Hom}(A,B)\neq\\{0\\}$ and
$\mathrm{Hom}(B,A)\neq\\{0\\}$ is guaranteed.

###### Proposition 2.26

Let $A$, $B$ be ISI-groups, where $\mathrm{Hom}(A,B)\neq\\{0\\}$,
$\mathrm{Hom}(B,A)\neq\\{0\\}$ and $\\{0\\}\neq H\leq A$, $\\{0\\}\neq F\leq
B$ are si-subgroups of $A$, $B$, respectively (in particular, $H=A$, $F=B$
provided $A$, $B$ are si-simple groups). Then the group $G=A\oplus B$ is an
ISI-group if, and only if, $\mathrm{Hom}(H,B)H=F$ and $\mathrm{Hom}(F,A)F=H$,
as moreover $\mathrm{Hom}(A,B)A=B$ and $\mathrm{Hom}(B,A)B=A$ provided both
$A$, $B$ are not si-simple.

Proof. ”$\Rightarrow$”. If, for a moment,

$U=\mathrm{Hom}(H,B)H\neq F,$

then $H\oplus U$ is a si-subgroup in $G$ with $H\oplus U\neq H\oplus F$ that
contradicts the uniqueness of the si-subgroup of an ISI-group. Thus,
hereafter, if both $A$ and $B$ are not si-simple, i.e., $H\neq A$ and $F\neq
B$ and, for instance, $\mathrm{Hom}(A,B)A=F$, then we have for the si-subgroup
$A\oplus F\neq H\oplus F$.

”$\Leftarrow$”. Assume that $\\{0\\}\neq U,V\leq G$ are some si-subgroups in
$G$. It suffices to show that $U=V$. To that goal, we have

$U=(U\cap A)\oplus(U\cap B),V=(V\cap A)\oplus(V\cap B).$

On this vein, since $U\cap A$, $V\cap A$ are si-subgroups in $A$ as well as
$U\cap B$, $V\cap B$ are si-subgroups in $B$, and $A$,$B$ are ISI-groups, one
has that $U\cap A=V\cap A$ and $U\cap B=V\cap B$ hold, provided that these
subgroups are not trivial.

Note that the condition $U\cap A=\\{0\\}$ is impossible. In fact, if we assume
in a way of contradiction that $U\cap A=\\{0\\}$, then $U\cap B\neq\\{0\\}$
since $U\neq\\{0\\}$. If, however, $U\cap B=F$, then

$\\{0\\}\neq H=\mathrm{Hom}(F,A)F\leq U\cap A,$

thus contradicting the equality $U\cap A=\\{0\\}$. But, if $U\cap B=B$, then
$U\cap A=\\{0\\}$, thus contradicting the condition
$\mathrm{Hom}(B,A)\neq\\{0\\}$. Analogously, one obtains that $U\cap
B\neq\\{0\\}$, $V\cap A\neq\\{0\\}$, $V\cap B\neq\\{0\\}$.

Assume now that $U\cap A=A$. Then, the condition
$\mathrm{Hom}(A,B)\neq\\{0\\}$ forces that $U\cap B\neq\\{0\\}$. So, we have
$U=A\oplus F$, where $F=U\cap B\neq B$, because $U\neq G$. In particular, $B$
is not si-simple.

Next, we consider the first case when $A$ is si-simple. To this purpose,
assume that $V\cap B=B$. Since $\mathrm{Hom}(B,A)\neq\\{0\\}$, it follows that
$V\cap A\neq\\{0\\}$ and so $V\cap A=A$ taking into account that $A$ is si-
simple. Therefore, it must be that $V=G$, thus contradicting the given
assumption on $V$. So, $V\cap B=F\neq B$ and since $V\cap A\neq\\{0\\}$, we
obtain that $V\cap A=A$, i.e., $V=A\oplus F$. Consequently, $U=V$ holds in the
case when $A$ is si-simple.

Finally, assume $A$ is not si-simple. If, however, $B$ is si-simple, then we
deduce that $U\cap B=B$ and so $U=G$ follows, a contradiction. So, the
condition $U\cap A=A$ follows, provided that $B$ is not si-simple. By
assumptions, we know that $\mathrm{Hom}(A,B)A=B$ and $\mathrm{Hom}(B,A)B=A$.
Thus, the equality $U\cap A=A$ assures that $U\cap B=B$, i.e., $U=G$, thus
contradicting $U\neq G$, as required.

As an immediate consequence, we extract the following:

###### Corollary 2.27

If $G$ is either a si-simple, s-irreducible torsion-free group or an ISI-
group, then, for every ordinal $m>1$, the group $G^{(m)}$ retains the same
properties.

Note that some other interesting results for si-subgroups and irreducible
groups are proved in [2, 3], respectively. We now will give some additions to
that as follows. Let us recall that, for a group $G$, the notation $\tau(G)$
means the set consisting of all types of the non-zero elements of $G$.

###### Proposition 2.28

If $G$ is a non-homogeneous torsion-free ISI-group, then
$\tau(G)=\\{t_{1},t_{2}\\}$, where $t_{1}<t_{2}$ and the non-trivial si-
subgroup of $G$ is exactly $G(t_{2})$.

Proof. We clearly have $\\{0\\}\neq G(t_{2})\neq G$ for some type $t_{2}$.
Since the si-subgroup is unique (see Proposition 2.10), the group $G(t_{2})$
is then homogeneous, i.e., the type $t_{2}$ is maximal. Letting
$t_{1}\in\tau(G)\setminus\\{t_{2}\\}$, it follows from $G(t_{1})\ncong
G(t_{2})$ that $G(t_{1})=G$. Consequently, we deduce that $t_{1}<t_{2}$ and
$\tau(G)=\\{t_{1},t_{2}\\}$, as stated.

Note that the si-subgroup $H=G(t_{2})$ from Proposition 2.28 is manifestly
s-irreducible. Indeed, if $\\{0\\}\neq F<H$ is a pure si-subgroup in $H$, then
we can plainly inspect that $\\{0\\}\neq\mathrm{Hom}(F,G)F\leq H$, so that
$F=H$, as claimed.

We are now recording the following consequence.

###### Corollary 2.29

If the torsion-free ISI-group $G$ has a non-$p$-pure si-subgroup $H$, then $G$
is homogeneous and, if $H$ is not si-simple, then $pG\leq F$ for every non-
trivial si-subgroup $F$ of $H$.

Proof. Arguing as above, we are aware that

$\sum_{\alpha\in\mathrm{Hom}(F,G)}\mathop{\rm Im}\nolimits\,\alpha=G,$

whence

$pG=\sum_{\alpha\in\mathrm{Hom}(F,G)}\mathop{\rm Im}\nolimits\,(p\alpha)\leq
F$

as $p\alpha\in\mathrm{Hom}(F,H)$ by exploiting Proposition 2.17.

Now, we are able to show the validity of the following two constructions.

###### Example 2.30

The next two points are true:

(i) For every type $\tau\neq(\infty,\infty,\dots)$, there exists a homogeneous
torsion-free group $G$ of type $\tau$ such that, for each natural number
$r>1$, $G$ has a si-subgroup of rank $\frac{(r+2)(r-1)}{2}$.

(ii) There exists a torsion-free group $G$ such that, for each natural number
$r\geq 2$, $G$ possesses a si-subgroup of rank $2r$.

Proof. (i) Let $G_{r}$ be the group from [10, §88, Example 5] having rank $r$
and set $G=\bigoplus_{r>1}G_{r}$. Then, the so-constructed group $G$ will have
type $(0,0,\dots)$. By virtue of [10, §88, Exersize 6], any subgroup in
$G_{r}$ of rank $\leq r-1$ is a free group, and hence it follows that the
following equality is true

$\mathrm{Hom}(\bigoplus_{2\leq i\leq r}G_{r},\bigoplus_{j>r}G_{j})=\\{0\\},$

because $G_{r}$ does not have unfree direct summands, i.e., each direct
component $\bigoplus_{2\leq i\leq r}G_{r}$ is a si-subgroup of $G$.

If now $R$ is a torsion-free group of rank $1$ and type
$\tau\neq(\infty,\infty,\dots)$, then one readily verifies that the standard
tensor product $G\otimes R$ over $\mathbb{Z}$ is a homogeneous group of type
$\tau$, equipped with the same properties as these of $G$, as needed.

(ii) Let $G_{r}$ be the group of rank $2r$ from [10, §88, Exersize 7] and put
$G=\bigoplus_{r\geq 2}G_{r}$. Then, since every subgroup in $G_{r}$ of rank
$\leq r$ is free and all factor-groups of $G_{r}$ of rank $\leq r$ are
divisible, one easily checks that each $G_{r}$ is a si-subgroup in $G$, as
required.

We shall now discover when ISI-groups are IFI-groups by arranging to prove the
following (as mentioned in the introductory section, the converse is always
true, because si-subgroups are fi-subgroups).

###### Proposition 2.31

(i) A torsion ISI-group $G$ is an IFI-group if, and only if, $G$ is a
$p$-group for some prime $p$ having the property $p^{2}G=\\{0\\}$ and,
moreover, if $pG\neq\\{0\\}$, then $\mathrm{rank}(G)=\mathrm{rank}(pG)$.

(ii) A mixed ISI-group need not be an IFI-group.

(iii) A torsion-free IFI-group is a si-simple group of idempotent type.

Proof. (i) It follows automatically from Theorems 2.5 and 2.11.

(ii) It follows directly from a combination of Theorem 2.5 (since there not
exists a mixed IFI-group) and Theorem 2.13 in which mixed ISI-groups are
described.

(iii) It follows immediately from the fact in Proposition 2.2 that torsion-
free IFI-groups are strongly IFI-groups, as in this case every si-subgroup is
isomorphic to the whole group and, consequently, all torsion-free IFI-groups
are themselves si-simple.

Note that si-simple groups are always ISI-groups since they have only trivial
si-subgroups. Also, there are si-simple groups that are not IFI-groups;
indeed, all separable torsion-free groups of a non-idempotent type are such
groups. However, a question which immediately arises is whether all si-simple
torsion-free groups of idempotent type are IFI-groups?

###  2.2 Strongly ISI-groups

We continue here with more details than the comments listed after Definition
6.

###### Lemma 2.32

The following conditions are equivalent:

(a) The group $G$ is a strongly ISI-group;

(b) The equality $\mathrm{Hom}(F,G)F=G$ holds for every subgroup $\\{0\\}\neq
F\leq G$;

(c) The group $G$ is a si-simple group.

Proof. (a) $\Rightarrow$ (c). Indeed, if $H\neq\\{0\\}$ is a si-subgroup, then
$H\cong G$, and if $f:H\rightarrow G$ is an isomorphism, then $f(H)=G\leq H$,
i.e., $H=G$, as needed.

The relationships (b) $\Leftrightarrow$ (c) and (c) $\Rightarrow$ (a) are
obvious, so we omit the details in proving them.

It is clear that a si-simple torsion-free group is a homogeneous group and
that every fi-simple group is si-simple. Moreover, in view of [1, Proposition
24], each fi-simple group is either $p$-elementary for some prime $p$ or is
divisible torsion-free.

Some other examples of si-simple groups are the following ones:

(1) Fully transitive homogeneous torsion-free groups of idempotent type are
si-simple (see, e.g., [1, Proposition 26]).

(2) For any type $\tau$, there exists a torsion-free si-simple group of
homogeneous type $\tau$. In fact, as such a group, it is possible to take any
separable homogeneous torsion-free group $G$ (see, for instance, [3,
Proposition 1(1)]).

(3) A non-zero torsion group $G$ is a si-simple group if, and only if, it is a
non-zero elementary $p$-group for some prime $p$. This claim follows at once
from [1, Proposition 24].

Furthermore, we give here some elementary but helpful observations:

$\bullet$ A torsion-free IFI-group is always a strongly ISI-group.

Indeed, we just need to combine Proposition 2.31 (iii) and Lemma 2.32 (c).

$\bullet$ A non-zero torsion group is a strongly ISI-group if, and only if, it
is a non-zero elementary $p$-group for some prime $p$.

Indeed, it suffices just to combine point (3) quoted above with Lemma 2.32
(c).

###  2.3 Weakly ISI-groups

Before starting our work, it is worth to notice that in [11], [12] and [13]
were examined those Abelian groups, both torsion and torsion-free, having
isomorphic proper fully invariant subgroup. Likewise, in [8] were investigated
in detail those Abelian groups in which all subgroups of infinite index are
free.

In this subsection, we will initiate the examination of the case when we have
a proper strongly invariant subgroup isomorphic to the whole group. For
simpleness of the exposition, we shall call these groups just weakly ISI-
groups. However, unfortunately, this class of groups is definitely not so
interesting, because each si-subgroup which is a weakly ISI-group coincides
with the whole group.

##  3 Concluding Discussion and Open Problems

We conclude this final section with some valuable comments on the obtained
results. In fact, summarizing all of what we have established so far, we can
just say that the properties of ISI-groups are totally different from these of
ISI-groups and IC-groups which were completely described in [4] and [7],
respectively. The primary reason for this discrepancy is that there is no
abundance of so many si-subgroups (compare also with [1]).

We close our work with six questions of interest and importance.

Problem 1. Does there exist s-irreducible that are not ISI-groups?

Problem 2. Is it true that si-subgroups of ISI-groups are also ISI-groups?

Problem 3. Does it follow that an s-irreducible torsion-free ISI-group is si-
simple?

Knowing with the aid of [4, Example 2.6] that there exists a homogeneous
torsion-free group of idempotent type and rank strictly greater that $1$ which
is not an IFI-group, it is reasonably logical to state the following.

Problem 4. Decide when a homogeneous torsion-free group of idempotent type and
rank $2$ is an ISI-group?

We know that each strongly invariant subgroup is always fully invariant, and
thus every IFI-group is an ISI-group, but the reverse implication fails. So,
in regard to Proposition 2.31 and the first bullet from Subsection 2.2, we may
ask the following.

Problem 5. Find suitable conditions under which each torsion-free ISI-group of
an idempotent type is an IFI-group.

In a way of similarity to Theorem 2.1 and Corollary 2.27, both listed above,
we may state our next query.

Problem 6. For a group $G$ of an idempotent type, does it follow that the
square $G\oplus G$ is an ISI-group if, and only if, $G\oplus G$ is an IFI-
group?

The restriction on the idempotent type is essential and cannot be ignored: in
fact, if $G$ is a torsion-free separable group of a non-idempotent type, then
it is necessarily si-simple and thus $G\oplus G$ is too si-simple and hence an
ISI-group but not an IFI-group, because $G\oplus G$ is still of a non-
idempotent type.

Acknowledgement. The authors would like to thank the unknown expert referee
for his/her valuable comments and suggestions which led to an improvement of
the article’s shape.

Funding: The scientific work of Andrey R. Chekhlov was supported by the
Ministry of Science and Higher Education of Russia (agreement No.
075-02-2023-943). The scientific work of Peter V. Danchev was supported in
part by the Bulgarian National Science Fund under Grant KP-06 No 32/1 of
December 07, 2019, as well as by the Junta de Andalucía, Grant FQM 264, and by
the BIDEB 2221 of TÜBÍTAK.

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