# Path partial groups

Antonio Díaz Ramos  Departamento de Álgebra, Geometría y Topología,
Universidad de Málaga, 29071-Málaga, Spain<EMAIL_ADDRESS>
ORCID: 0000-0002-1669-1374 ,  Rémi Molinier  Institut Fourier, UMR 5582,
Laboratoire de Mathématiques,
Université Grenoble Alpes, CS 40700, 38058 Grenoble cedex 9, France
<EMAIL_ADDRESS>
ORCID: 0000-0002-3742-5307 and  Antonio Viruel  Departamento de Álgebra,
Geometría y Topología, Universidad de Málaga, 29071-Málaga, Spain
<EMAIL_ADDRESS>
ORCID: 0000-0002-1605-5845

###### Abstract.

It is well known that not every finite group arises as the full automorphism
group of some group. Here we show that the situation is dramatically different
when considering the category of partial groups, ${{\mathcal{P}}art}$, as
defined by Chermak: given any group $H$ there exists infinitely many non
isomorphic partial groups ${\mathbb{M}}$ such that
$\operatorname{Aut}_{{\mathcal{P}}art}({\mathbb{M}})\cong H$. To prove this
result, given any simple undirected graph $G$ we construct a partial group
${\mathbb{P}}(G)$, called the path partial group associated to $G$, such that
$\operatorname{Aut}_{{\mathcal{P}}art}\big{(}{\mathbb{P}}(G)\big{)}\cong\operatorname{Aut}_{{\mathcal{G}}raphs}(G)$.
As a consequence, we solve the Kahn realizability problem in the category of
simplicial sets.

###### Key words and phrases:

Partial group, automorphism, graph

###### 2020 Mathematics Subject Classification:

Primary: 20F29; Secondary: 20N99, 55U10, 55R35

The first and third authors were partially supported by Ministerio de Ciencia
e Innovación (Spain) grant PID2020-118753GB-I00

##  1\. Introduction

From the seminal work of Galois to the discovery of sporadic simple groups,
describing abstract groups as the full automorphisms group of a mathematical
object can be considered the source of Group Theory, and it certainly sits in
the core of Representation Theory. Within this framework, it is then natural
to ask whether for a fixed category $\mathcal{C}$, and given an abstract group
$H$, there exists an object $X$ in $\mathcal{C}$ such that
$H\cong\operatorname{Aut}_{\mathcal{C}}(X)$. If any group (resp. finite group)
$H$ can be so represented, the category $\mathcal{C}$ is said to be universal
(resp. finitely universal) [4, Section 4.1].

Some categories have already been shown to be either universal or finitely
universal. For example, ${{\mathcal{G}}raphs}$, the category of simple
undirected graphs, is known to be universal [22, 19, 27], and this has been
used to show that algebras over any field are finitely universal [13]. In
addition, the homotopy category of rational elliptic spaces is universal [16],
and the category of compact hyperbolic $n$-manifolds and their isometries is
finitely universal [5] for any $n\geq 2$. However, in general identifying
universal categories is a hard task and, for instance, deciding whether Galois
extensions over the field of rational numbers is finitely universal, the
Inverse Problem of Galois Theory [29], is still an open question. A more
comprehensive survey of this topic may be found in [4] and [24, Introduction].

Ironically, the category ${{\mathcal{G}}roups}$ is not even finitely
universal; it is well known that a nontrivial cyclic group of odd order cannot
arise as the full automorphism group of any group. So it is natural to ask
whether it is possible to “sensibly enlarge” the category
${{\mathcal{G}}roups}$ to a universal category $\mathcal{C}$. More precisely

######  Question 1.1.

Does there exist a universal category $\mathcal{C}$ such that
${{\mathcal{G}}roups}$ is fully embedded in $\mathcal{C}$?

We tackle this question by considering ${{\mathcal{P}}art}$, the category of
partial groups (see Section 2 for definitions). Partial groups, defined by
Chermak [9, Section 2], generalise the concept of group and are introduced as
a setting for the study of the $p$-local structure of finite groups within the
framework of fusions systems as defined by Broto-Levi-Oliver [8] (see also the
monographies [2] and [18]). Every group gives rise to a partial group in a
natural way [9, Example 2.4.(1)] and so ${{\mathcal{G}}roups}$ is fully
embedded in ${{\mathcal{P}}art}$. Therefore we answer Question 1.1 in the
positive by proving

######  Theorem A (Theorem 5.3).

The category ${{\mathcal{P}}art}$ is universal. Moreover, given an abstract
group $H$ there exist infinitely many non isomorphic partial groups
${\mathbb{M}}$ such that
$\operatorname{Aut}_{{\mathcal{P}}art}({\mathbb{M}})\cong H$.

In categories where a notion of homotopy equivalence between maps exists, a
related question is whether any given abstract group may be realized as the
group of homotopy classes of self-homotopy equivalences of some object of the
category. This problem, that is usually known as the Kahn realizability
problem, has been considered in the category of $CW$-complexes in the
aforementioned work [16] and in [17, 14, 15, 6], as well as in the category of
finite topological spaces in [11]. Using Theorem A, we solve the Kahn
realizability problem in the category of simplicial sets.

######  Theorem B (Theorem 6.1).

Given an abstract group $H$, there exist infinitely many non-homotopy
equivalent simplicial sets ${\mathcal{K}}$ such that its group of homotopy
classes of pointed simplicial self-homotopy equivalences is isomorphic to $H$.

The proof of Theorem A is built upon a functor, defined in Section 3, that
maps every pair $(G,\mathcal{H})$, for $G=(V,E)$ a simple undirected graph and
$\mathcal{H}=\\{H_{v}\\}_{v\in V}$ a collection of non-trivial groups, to a
partial group ${\mathbb{M}}(G,\mathcal{H})$ such that the group of
automorphisms
$\operatorname{Aut}_{{\mathcal{P}}art}\big{(}{\mathbb{M}}(G,\mathcal{H})\big{)}$
is closely related to the groups $\operatorname{Aut}_{{\mathcal{G}}raphs}(G)$
and $\prod_{v\in V}\operatorname{Aut}_{{\mathcal{G}}roups}(H_{v})$ (see
Theorem 4.4). The algebraic structure of the partial group
${\mathbb{M}}(G,\mathcal{H})$ is related to path concatenation within $G$. In
fact, the special case with $\mathcal{H}$ consisting of copies of
${\mathbb{Z}}_{2}$ is the central character in this work: path partial groups.

######  Definition 1.2.

Let $G$ be a simple undirected graph. The _path partial group_ associated to
the graph $G$, denoted by ${\mathbb{P}}(G)$, is the partial group
${\mathbb{M}}(G,\\{{\mathbb{Z}}_{2}\\}_{v\in V}).$

The algebraic structure of ${\mathbb{P}}(G)$ can be easily described in terms
of paths in $G$, and it resembles the ideas underlying path algebras, i.e.,
the algebra associated to a quiver [3, II.1.2]. The following result is the
key ingredient in the proof of Theorem A.

######  Theorem C (Theorem 5.2).

Let $G$ be a simple undirected graph. Then

$\operatorname{Aut}_{{\mathcal{P}}art}\big{(}{\mathbb{P}}(G)\big{)}\cong\operatorname{Aut}_{{\mathcal{G}}raphs}(G).$

In general, the graph $G$ can be recovered up to isomorphism from the partial
group ${\mathbb{M}}(G,\mathcal{H})$ whenever the collection $\mathcal{H}$
consists of locally finite groups (see Theorem 7.7). In particular, the path
partial group associated to a graph $G$ determines the isomorphism type of
$G$, that is, the path partial group is a classifying invariant for graphs.
The situation for path algebras is similar: a quiver can be recovered up to
isomorphism from its path algebra [3, II.3.6] and, in addition, some
invariants of the quiver correspond to invariants of its associated path
algebra [3, II]. Therefore, it makes sense to propose the following problem.

######  Problem 1.3.

Describe graph theoretical invariants of a simple undirected graph in terms of
algebraic invariants of its associated path partial group.

Besides the aforementioned setup of quivers and their associated path
algebras, the kind of study proposed in Problem 1.3 is well established in
other fields as directed graphs and their associated Leavitt path algebras,
see [1] or the Simplicity Theorem [1, 2.9.1] as a concrete instance of
interaction between the graph side and its algebraic counterpart. In other
areas as finite simplicial graphs and their associated right-angled Artin
groups, studying this type of relations is also an active line of work, see
[21, Problem 1.1]. In that work, the authors add the graph theoretical
property of $k$-colorability to the list of properties of simplical graphs
that can read off from the algebraic properties of its associated right-angled
Artin group. This list includes, among others, the graph theoretical
properties of being a join or disconnected, and these properties correspond to
the algebraic properties of decomposing as a non-trivial direct product or
decomposing as a non-trivial free product respectively. See [21, p. 2] for
more details.

### Outline of the paper:

We start giving in Section 2 basic notions about partial groups and some
related constructions as colimit of partial groups. This preliminaries are the
necessary background to construct the partial group
${\mathbb{M}}(G,\mathcal{H})$ associated to a decorated simple undirected
graph $(G,\mathcal{H})$ in Section 3. Then in Section 4 we deepen into this
construction and show how to embed the category of decorated graphs into the
category of partial groups. In Section 5, we restrict the study to path
partial groups and prove Theorems A and C. We also include in this section a
digress on rigidity in the categories of groups, partial groups and
topological spaces. The Kahn realizability problem is discussed in Section 6,
where Theorem B is also proven. Finally, in Section 7, we begin the study of
Problem 1.3.

### Acknowledgements:

The authors are thankful to Ellen Henke for showing interest in these results
and pointing out reference [28], and to Edoardo Salati for providing us with
an earlier version of his work. The authors also want to thank Carles Broto
for introducing us to the arguments in [7] that lead to Theorem B.

##  2\. Basics on partial groups

The notion of partial group is due to Chermak. Here we introduce the basic
definitions and some useful properties that are needed in the following
sections. More details on the subject can be found in [9, Section 2] or in the
preprint [10, Section 1].

For $X$ a set, $\mathbb{W}(X)$ will denote the free monoid on $X$ and for two
words $u,v\in\mathbb{W}(X)$, $u\circ v$ will denote the concatenation of $u$
and $v$. We also identify $X$ as the subset of words of length 1 in
$\mathbb{W}(X)$. Finally, given two sets $X$ and $Y$ and a map $\varphi\colon
X\to Y$, we will denote by
$\overline{\varphi}\colon\mathbb{W}(X)\to\mathbb{W}(Y)$ the map induced by
$\varphi$ defined as follows: for
$u=(x_{1},x_{2},\dots,x_{n})\in\mathbb{W}(X)$,
$\overline{\varphi}(u)=(\varphi(x_{1}),\varphi(x_{2}),\dots,\varphi(x_{n}))$.

######  Definition 2.1.

Let ${\mathcal{M}}$ be a set and
$\mathbb{D}\subseteq\mathbb{W}({\mathcal{M}})$ be a subset such that,

  1. (D1)

${\mathcal{M}}\subseteq\mathbb{D}$; and

  2. (D2)

$u\circ v\in\mathbb{D}\Rightarrow u,v\in\mathbb{D}$ (in particular,
$\emptyset\in\mathbb{D}$).

A mapping $\Pi:\mathbb{D}\rightarrow{\mathcal{M}}$ is a _product_ if

  1. (P1)

$\Pi$ restricts to the identity on ${\mathcal{M}}$; and

  2. (P2)

if $u\circ v\circ w\in\mathbb{D}$ then $u\circ\Pi(v)\circ w\in\mathbb{D}$ and

$\Pi(u\circ v\circ w)=\Pi\left(u\circ\Pi(v)\circ w\right).$

The _unit_ of $\Pi$ is then defined as $\Pi(\emptyset)$ that we will denote
$1_{\mathcal{M}}$ or $1$ when there is no ambiguities. A _partial monoid_ is a
triple $({\mathcal{M}},\mathbb{D},\Pi)$ where $\Pi$ is a product defined on
$\mathbb{D}$ and $\mathbb{D}$ is called the _domain_ of
$({\mathcal{M}},\mathbb{D},\Pi)$.

An _inversion_ on ${\mathcal{M}}$ is an involutory bijection $x\mapsto x^{-1}$
on ${\mathcal{M}}$ together with the induced mapping $u\mapsto u^{-1}$ on
$\mathbb{W}({\mathcal{M}})$ defined by,

$u=(x_{1},x_{2},\dots,x_{n})\mapsto(x_{n}^{-1},x_{n-1}^{-1},\dots,x_{1}^{-1}).$

A _partial group_ is a tuple
${\mathbb{M}}=\left({\mathcal{M}},\mathbb{D},\Pi,(-)^{-1}\right)$ where
$\left({\mathcal{M}},\mathbb{D},\Pi\right)$ is a partial monoid and $(-)^{-1}$
is an inversion on ${\mathcal{M}}$ satisfying

  1. (P3)

if $u\in\mathbb{D}$ then $(u^{-1},u)\in\mathbb{D}$ and $\Pi(u^{-1}\circ u)=1$.

A word $w\in\mathbb{D}$ will be called _non-degenerate_ if there is no $1$ in
$w$. Then, $\mathbb{D}$ is totally determined by its non-degenerated words.

######  Remark 2.2.

The notion of locality [9, Definition 2.9], a particular type of partial
group, plays a central role in Chermak’s work and the study of $p$-local
structure of finite groups. One can check that the partial groups considered
in this work (Definitions 3.4 and 3.6) are not localities in general as they
already fail to be _objective_ partial groups ([9, Definition 2.6]).

######  Example 2.3.

Let ${\mathbb{M}}=\left({\mathcal{M}},\mathbb{D},\Pi,(-)^{-1}\right)$ be a
partial group. If $\mathbb{D}=\mathbb{W}({\mathcal{M}})$ then ${\mathbb{M}}$
is a group via the binary operation
$(x,y)\in{\mathcal{M}}^{2}\mapsto\Pi(x,y)\in{\mathcal{M}}$. Moreover, if $H$
is a group and $\Pi\colon\mathbb{W}(H)\to H$ is the multivariate product
induced by the binary product on $H$, $(H,\mathbb{W}(H),\Pi,(-)^{-1})$ is a
partial group.

######  Example 2.4.

Let $\mathcal{F}(a)=\\{1,a,a^{-1}\\}$ and set the non-degenerated words of
$\mathbb{D}_{a}$ to be all possible words alternating $a$’s and $a^{-1}$’s. In
other words, the non-degenerated words of $\mathbb{D}_{a}$ are all the
different finite sub-words of the infinite word
$(a,a^{-1},a,a^{-1},a,a^{-1},\dots)$. The inversion is understood and, for any
word $u\in\mathbb{D}\left(\mathcal{F}(a)\right)$,

$\Pi(u)=\begin{cases}1&\text{if the number of $a$'s equal the number of
$a^{-1}$'s},\\\ a&\text{if the number of $a$'s exceed the number of $a^{-1}$'s
(necessarily by 1)},\\\ a^{-1}&\text{if the number of $a^{-1}$'s exceed the
number of $a$'s (necessarily by 1)}.\end{cases}$

One can then check that
$\left(\mathcal{F}(a),\mathbb{D}_{a},\Pi_{a},(-)^{-1}\right)$ defines a
partial group.

This last example is actually the _free partial group on the set $\\{a\\}$_ as
it is detailed in [10, Lemma 1.12].

######  Definition 2.5.

Let ${\mathbb{M}}=\left({\mathcal{M}},\mathbb{D},\Pi,(-)^{-1}\right)$ be a
partial group. A _partial subgroup_ of ${\mathbb{M}}$ is a subset
${\mathcal{N}}$ of ${\mathcal{M}}$ closed by inversion and such that
$\Pi\left(\mathbb{D}\cap\mathbb{W}({\mathcal{N}})\right)\subseteq{\mathcal{N}}$.
Its partial group structure is given by
$\left({\mathcal{N}},\mathbb{D}\cap\mathbb{W}({\mathcal{N}}),\Pi,(-)^{-1}\right)$.
If $\mathbb{D}\cap\mathbb{W}({\mathcal{N}})=\mathbb{W}({\mathcal{N}})$ then we
say that ${\mathcal{N}}$ is a _subgroup_ of ${\mathbb{M}}$.

######  Definition 2.6.

Let
${\mathbb{M}}_{1}=\left({\mathcal{M}}_{1},\mathbb{D}_{1},\Pi_{1},(-)^{-1}\right)$
and
${\mathbb{M}}_{2}=\left({\mathcal{M}}_{2},\mathbb{D}_{2},\Pi_{2},(-)^{-1}\right)$
be partial groups. A map of sets
$\varphi\colon{\mathcal{M}}_{1}\to{\mathcal{M}}_{2}$ is a _homomorphism of
partial groups_ from ${\mathbb{M}}_{1}$ to ${\mathbb{M}}_{2}$ if

  1. (H1)

$\overline{\varphi}\left(\mathbb{D}_{1}\right)\subseteq\mathbb{D}_{2}$;

  2. (H2)

for any $u\in\mathbb{D}_{1}$,
$\Pi_{2}\left(\overline{\varphi}(u)\right)=\varphi\left(\Pi_{1}(u)\right)$.

The _kernel_ of $\varphi$ is the partial subgroup of ${\mathcal{M}}_{1}$ given
by

$\ker(\varphi)=\\{x\in{\mathcal{M}}_{1}\text{ $|$
}\varphi(x)=1_{{\mathcal{M}}_{2}}\\}.$

An homomorphism $\varphi\colon{\mathcal{M}}_{1}\to{\mathcal{M}}_{2}$ is an
_isomorphism_ of partial groups if the map $\varphi$ is bijective and
$\varphi^{-1}$ is also a morphism of partial groups. Finally, an
_automorphism_ of the partial group ${\mathcal{M}}_{1}$ is an _isomorphism_
$\varphi\colon{\mathcal{M}}_{1}\to{\mathcal{M}}_{1}$.

######  Lemma 2.7 ([10, Lemma 1.15]).

Consider partial groups
${\mathbb{M}}_{1}=\left({\mathcal{M}}_{1},\mathbb{D}_{1},\Pi_{1},(-)^{-1}\right)$
and
${\mathbb{M}}_{2}=\left({\mathcal{M}}_{2},\mathbb{D}_{2},\Pi_{2},(-)^{-1}\right)$
and a homomorphism of partial groups between them,
$\varphi\colon{\mathcal{M}}_{1}\to{\mathcal{M}}_{2}$. Then, if ${\mathcal{N}}$
is a subgroup of ${\mathbb{M}}_{1}$, $\varphi({\mathcal{N}})$ is a subgroup of
${\mathbb{M}}_{2}$.

The category with objects the partial groups and equipped with the notion of
homomorphism above and the usual composition of maps is denoted
${{\mathcal{P}}art}$. In particular, given a partial group ${\mathcal{M}}$,
the set of all its automorphisms is a group that we will denote
$\operatorname{Aut}_{{\mathcal{P}}art}({\mathcal{M}})$. Notice that
${{\mathcal{P}}art}$ contains the category ${{\mathcal{G}}roups}$ of groups as
a full subcategory as illustrated in Example 2.3. We are here interested in
the following question.

######  Question 2.8.

Is the category ${{\mathcal{P}}art}$ universal? That is, given an arbitrary
group $H$, does there exist a partial group ${\mathbb{M}}$ such that
$\operatorname{Aut}_{{\mathcal{P}}art}({\mathbb{M}})=H$?

The category ${{\mathcal{P}}art}$ has all limits and all colimits, and here we
only discuss colimits as we will employ them later. See [10, Theorem 1.17],
and [28] for a correct version of the construction of limits and colimits. So
let $\mathcal{C}$ be a small category and let
${\mathbb{M}}\colon\mathcal{C}\to{{\mathcal{P}}art}$ be a functor,
$C\mapsto{\mathbb{M}}(C)=({\mathcal{M}}(C),\mathbb{D}(C),\Pi(C),(-)_{C}^{-1})$.
Then the colimit in the category ${{\mathcal{P}}art}$,

$\mathop{colim}\limits_{C\in\mathcal{C}}{\mathbb{M}}(C)=({\mathcal{M}},\mathbb{D},\Pi,(-)^{-1}),$

has alphabet ${\mathcal{M}}$ equal to the quotient set,

${\mathcal{M}}=\cup_{C\in\mathcal{C}}{\mathcal{M}}(C)/\sim$

where $\sim$ is the smallest equivalence relation that contains the following
relation,

(C1)
$f\in\operatorname{Mor}_{\mathcal{C}}(C_{1},C_{2}),x\in{\mathcal{M}}(C_{1})\Rightarrow
x\sim{\mathcal{M}}(f)(x),$

and such that there is a map $\Pi$ closing the following diagram, where
$\mathbb{D}$ is defined as the subset of words $u$ in the free monoid
$\mathbb{W}({\mathcal{M}})$ for which there exists an object
$C\in\operatorname{Ob}(\mathcal{C})$ and a word $v$ in $\mathbb{D}(C)$ such
that $u$ is the image of $v$ under the component-wise application of $\sim$,
the rightmost vertical arrow is induced by $\sim$, and the leftmost vertical
arrow by its component-wise application,

(C2) $\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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48.75218pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\cup_{C\in\mathcal{C}}\Pi(C)}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 108.06332pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 108.06332pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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We denote the equivalence classes corresponding to $\sim$ with brackets,
$[-]$. Inversion on ${\mathcal{M}}$ is the only map making commutative the
following diagram, where again vertical arrows are induced by $\sim$,

$\textstyle{\cup_{C\in\mathcal{C}}{\mathcal{M}}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup_{C\in\mathcal{C}}(-)_{C}^{-1}}$$\textstyle{\cup_{C\in
C}{\mathcal{M}}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)^{-1}}$$\textstyle{{\mathcal{M}}.}$

##  3\. Partial groups out of simple graphs

Our goal in this section is to construct a partial group out of a graph
decorated with a group on each vertex. Given a collection of groups
$H_{1},\dots,H_{n}$, we start considering a free product
$H=\ast_{i=1}^{n}H_{i}$. If we set $H_{i}^{*}=H_{i}\smallsetminus\\{1\\}$,
then the group $H$ itself may be identified with the subset of words in
$\mathbb{W}(H_{1}^{*}\cup H_{2}^{*}\cup\cdots\cup H_{n}^{*})$ which are
_reduced_.

######  Definition 3.1.

A word $u=(g_{1},g_{2},\dots,g_{r})\in\mathbb{W}(H_{1}^{*}\cup
H_{2}^{*}\cup\cdots\cup H_{n}^{*})$ is _reduced_ if it is the empty word or if
$g_{j}\in H_{i_{j}}$ then $i_{j}\neq i_{j+1}$, $j=1,\ldots,r-1$. Given a
reduced word $u=(g_{1},g_{2},\dots,g_{r})\in\mathbb{W}(H_{1}^{*}\cup
H_{2}^{*}\cup\cdots\cup H_{n}^{*})$, its _length_ is defined as $|u|=r$.

By [12, Theorem 21], there is a map

($\Pi$) $\Pi\colon\mathbb{W}(H_{1}^{*}\cup H_{2}^{*}\cup\cdots\cup
H_{n}^{*})\to\mathbb{W}(H_{1}^{*}\cup H_{2}^{*}\cup\cdots\cup H_{n}^{*})$

that takes any word $w$ to its unique reduced form $\Pi(w)$. We will employ
this notation throughout this section. The product in $H$ corresponds then to
concatenation $\circ$ followed by $\Pi$.

######  Definition 3.2.

A word $u=(g_{1},g_{2},\dots,g_{r})\in\mathbb{W}(H_{1}^{*}\cup
H_{2}^{*}\cup\cdots\cup H_{n}^{*})$ is _cyclically reduced_ if all of its
cyclic permutations
$(g_{2},g_{3},\ldots,g_{r},g_{1}),\ldots,(g_{r},g_{1},\ldots,g_{r-1})$ are
reduced [12, p. 34].

The inverse of a reduced or cyclically reduced word is also a reduced or
cyclically reduced word respectively. One may also notice that if $u$ is a
reduced word, then $|\Pi(u\circ u)|\leq 2|u|$ and equality holds if and only
if $u$ is cyclically reduced.

######  Definition 3.3.

The category ${{\mathcal{G}}raphs}$ is the category with objects the simple
undirected graphs and morphisms the graphs homomorphisms.

In what follows, $G=(V,E)$ is an object in ${{\mathcal{G}}raphs}$ and
$\mathcal{H}=\\{H_{v}\\}_{v\in V}$ is a collection of groups. Our aim is to
define a partial group depending on $G$ and $\mathcal{H}$. Now, for a clique
$C\subset V$ in $G$, we consider the free product
$H_{C}=\mathop{\ast}\limits_{v\in C}H_{v}$ and the following construction.

######  Definition 3.4.

For a clique $C$ in $G$, consider the $4$-tuple,

${\mathbb{M}}(C)=(\operatorname{CR}(C),\mathbb{D}(C),\Pi(C),(-)_{C}^{-1}),$

where,

  1. (1)

$\operatorname{CR}(C)$ is the set of cyclically reduced words in
$\mathbb{W}(H_{1}^{*}\cup H_{2}^{*}\cup\cdots\cup H_{n}^{*})$,

  2. (2)

$\mathbb{D}(C)$ is the set of words
$w=(u_{1},u_{2},\dots,u_{n})\in\mathbb{W}(\operatorname{CR}(C))$ such that,

(H) $\text{for all }1\leq i\leq j\leq n,\qquad\Pi(u_{i}\circ
u_{i+1}\circ\cdots\circ u_{j})\in\operatorname{CR}(C),$

  3. (3)

$\Pi(C)\colon\mathbb{D}(C)\to\operatorname{CR}(C)$ is given by reduced form,

$(u_{1},u_{2},\dots,u_{n})\in\mathbb{D}(\operatorname{CR}(C))\mapsto\Pi(u_{1}\circ
u_{2}\circ\cdots\circ u_{n})\in\operatorname{CR}(C)\text{, and}$

  4. (4)

inversion $(-)_{C}^{-1}\colon\operatorname{CR}(C)\to\operatorname{CR}(C)$ is
given by inversion in the group $H_{C}$ restricted to $\operatorname{CR}(C)$.

Although the graph $G$ may have infinite order, the words we are considering
here are always of finite length, and thus involve only a finite number of
vertices. Therefore everything here is well defined.

######  Theorem 3.5.

Let $G=(V,E)$ be a simple undirected graph, $\mathcal{H}=\\{H_{v}\\}_{v\in V}$
a collection of groups, and $C$ a clique in $G$. Then ${\mathbb{M}}(C)$ is a
partial group.

###### Proof.

By construction, (D1), (D2) are both satisfied. Since elements in
${\mathcal{M}}(C)$ are cyclically reduced words, they are equal to their
reduced forms and (P1) is satisfied. The axiom (P2) is satisfied thanks to
(H). Finally, to check (P3), let $w=(u_{1},u_{2},\dots,u_{n})\in\mathbb{D}(C)$
and set

$(v_{1},v_{2},\dots,v_{2n}):=(w,w^{-1})=(u_{1},u_{2},\dots,u_{n},u_{n}^{-1},u_{n-1}^{-1},\dots,u_{1}).$

Choose $1\leq i\leq j\leq 2n$ and let $\omega=(v_{i},v_{i+1},\dots,v_{j})$.
Then, four distinct situations may happen,

  1. (a)

$i\leq j\leq n$ and then $\omega$ is a subword of $w$;

  2. (b)

$n+1\leq i\leq j$ and then $\omega$ is a subword of $w^{-1}$;

  3. (c)

$i\leq n<j$ and $n-i+1\geq j-n$ and then
$\Pi(\omega)=\Pi(u_{i},u_{i+1},\dots,u_{2n-j})$;

  4. (d)

$i\leq n<j$ and $n-i+1\leq j-n$ and then
$\Pi(\omega)=\Pi(u_{i-1}^{-1},\dots,u_{2n+1-j}^{-1})$.

Therefore, in all cases $\Pi(\omega)=\Pi(\alpha)$ where $\alpha$ is a subword
of either $w$ or $w^{-1}$, and since both $w,w^{-1}\in\mathbb{D}(C)$, (P3) is
satisfied by condition (H). ∎

Let $\Delta$ be the poset of cliques $C\subset V$ in $G$. An inclusion
$C_{1}\subset C_{2}$ between cliques gives rise to an inclusion map
$\operatorname{CR}(C_{1})\subset\operatorname{CR}(C_{2})$ and to its
associated map of partial groups ${\mathbb{M}}(C_{1})\to{\mathbb{M}}(C_{2})$.
This way, we have a functor
${\mathbb{M}}(-)\colon\Delta\to{{\mathcal{P}}art}$.

######  Definition 3.6.

Let $G=(V,E)$ be a simple undirected graph and $\mathcal{H}=\\{H_{v}\\}_{v\in
V}$ a collection of groups. We define the partial group associated to the pair
$(G,\mathcal{H})$ by

${\mathbb{M}}(G,\mathcal{H})=({\mathcal{M}}(G,\mathcal{H}),\mathbb{D}(G,\mathcal{H}),\Pi(G,\mathcal{H}),(-)_{G,\mathcal{H}}^{-1}):=\mathop{colim}\limits_{C\in\Delta}{\mathbb{M}}(C).$

Because of the simple shape of the category $\Delta$ and the functor
${\mathbb{M}}(-)$, it is straightforward that, in Definition 3.6, Equation
(C1) suffices to define the equivalence relation $\sim$ for the colimit. In
fact, we define ${\mathcal{M}}(G,\mathcal{H})$ using only that condition. Then
the map

(S) $\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
31.08371pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{M}}(G,\mathcal{H})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
55.08371pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
55.08371pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\\{\text{ cyclically reduced words in
}\mathbb{W}(\cup_{v\in V}H_{v}^{*})\text{
}\\}}$}}}}}}}{\hbox{\kern-31.08371pt\raise-21.31804pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{[(g_{1},g_{2},\ldots,g_{r})]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
31.08371pt\raise-21.31804pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
122.54466pt\raise-21.31804pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
122.54466pt\raise-21.31804pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(g_{1},g_{2},\ldots,g_{r})}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

is well defined and injective, where
$(g_{1},g_{2},\ldots,g_{r})\in\operatorname{CR}(C)$ for some clique $C$.
Moreover, this map together with uniqueness of the reduced form ($\Pi$),
ensure that condition (C2) is satisfied too. From now on, we will make the
identification given by (S) and the corresponding one for
$\mathbb{D}(G,\mathcal{H})$ without further notice.

######  Lemma 3.7.

Let $G=(V,E)$ be a simple undirected graph and $\mathcal{H}=\\{H_{v}\\}_{v\in
V}$ be a collection of groups.

  1. (a)

For all $v\in V$, $H_{v}$ is a subgroup of ${\mathbb{M}}(G,\mathcal{H})$.

  2. (b)

If $K\subseteq{\mathcal{M}}(G,\mathcal{H})$ is a non-trivial finite subgroup
of ${\mathbb{M}}(G,\mathcal{H})$, then there exists a unique $v\in V$ such
that $K\leq H_{v}$.

  3. (c)

For all $v,v^{\prime}\in V$ $v\neq v^{\prime}$, and all
$(h_{v},h_{v^{\prime}})\in H_{v}^{*}\times H_{v^{\prime}}^{*}$,
$((h_{v}),(h_{v^{\prime}}))\in\mathbb{D}(G,\mathcal{H})$ if and only if
$\\{v,v^{\prime}\\}\in E$.

###### Proof.

Property (a) is a consequence of Equation (S). For point (b), consider a
cyclically reduced word $u\in K\smallsetminus\\{1\\}$. If length $|u|>1$, then
the length of the cyclically reduced word

$\Pi(\overbrace{u\circ u\ldots\circ u}^{\text{$n$ times}})$

is $n|u|$. Therefore, if $u\in{\mathcal{M}}(G,\mathcal{H})$ is an element of
finite order, $u$ is a word of length 1 and it is an element of one of the
groups $H_{v}$ for $v\in V$. Now, assume there exist $k,k^{\prime}\in
K\smallsetminus\\{1\\}$ and $v,v^{\prime}\in V$ such that $k\in H_{v}^{*}$,
$k^{\prime}\in H_{v^{\prime}}^{*}$ but $v\neq v^{\prime}$. Then
$(k,k^{\prime})$ is a cyclically reduced word of length 2 in
$\mathbb{W}(\cup_{v\in V}H_{v}^{*})$ and, as $K$ is a subgroup of
${\mathcal{M}}(G,\mathcal{H})$, it belongs to $K$ and
${\mathcal{M}}(G,\mathcal{H})$. In particular, it cannot be an element of
finite order in ${\mathcal{M}}(G,\mathcal{H})$.

For part (c), assume first that
$((h_{v}),(h_{v^{\prime}}))\in\mathbb{D}(G,\mathcal{H})$. Then, by the
definition of colimit of partial groups, there exists a clique $C$ of $G$ and
a word $(u_{1},u_{2})\in\mathbb{D}(C)$ such that
$((h_{v}),(h_{v^{\prime}}))=(u_{1},u_{2})$ with
$u_{i}\in\operatorname{CR}(C)$. Thus, we must have $u_{1}=(h_{v})$ and $v\in
C$, and $u_{2}=(h_{v^{\prime}})$ and $v^{\prime}\in C$. In particular,
$\\{v,v^{\prime}\\}\in E$. The reversed implication is straightforward. ∎

Now we define the category of decorated graphs, which is built on top of the
category ${{\mathcal{G}}raphs}$.

######  Definition 3.8.

The category ${{\mathcal{D}}ecGraphs}$ is the category with object set the
pairs $(G,\mathcal{H})$, where $G=(V,E)\in Ob({{\mathcal{G}}raphs})$ and
$\mathcal{H}=\\{H_{v}\\}_{v\in V}$ is a collection of groups indexed by the
vertices of $G$, and with morphisms the pairs

$(f_{G},f_{\mathcal{H}})\colon(G,\mathcal{H})\to(G^{\prime},\mathcal{H}^{\prime}),$

where $f_{G}\in\operatorname{Mor}_{{\mathcal{G}}raphs}(G,G^{\prime})$ and
$f_{\mathcal{H}}=\\{f_{v}\\}_{v\in V}$ is a collection of _injective_ group
homomorphisms $f_{v}\colon H_{v}\to H^{\prime}_{f_{G}(v)}$ for all vertices
$v$ of $G$.

A morphism $(f_{G},f_{\mathcal{H}})$ in ${{\mathcal{D}}ecGraphs}$ takes the
clique $C=\\{v_{1},\ldots,v_{n}\\}$ in $G$ to the clique
$C^{\prime}=\\{f_{G}(v_{1}),\ldots,f_{G}(v_{n})\\}$ in $G^{\prime}$, and note
that the restriction ${f_{G}}_{|C}$ is injective. In turn, it takes the
cyclically reduced word

$u=(g_{1},g_{2},\dots,g_{r})\in\operatorname{CR}(C)\text{ with $g_{j}\in
H_{v_{i_{j}}}$,}$

to the cyclically reduced word

$u^{\prime}=(f_{v_{i_{1}}}(g_{1}),f_{v_{i_{2}}}(g_{2}),\dots,f_{v_{i_{r}}}(g_{r}))\in\operatorname{CR}(C^{\prime})\text{
with $f_{v_{i_{j}}}(g_{j})\in H^{\prime}_{f_{G}(v_{i_{j}})}$.}$

To ensure that $u^{\prime}$ is a cyclically reduced word we do need that the
morphisms $f_{v_{i_{1}}},f_{v_{i_{2}}},\ldots,f_{v_{i_{r}}}$ are injective
group homomorphisms. It is easy to check that the map defined above,

(V)
$\varphi_{(f_{G},f_{\mathcal{H}})}\colon\operatorname{CR}(C)\to\operatorname{CR}(C^{\prime}),$

induces a homomorphism of partial groups from ${\mathbb{M}}(C)$ to
${\mathbb{M}}(C^{\prime})$. Moreover, we have a natural transformation between
the functors

${\mathbb{M}}(-)\colon\Delta\to{{\mathcal{P}}art}\qquad\text{and}\qquad{\mathbb{M}}(-)^{\prime}\colon\Delta^{\prime}\to{{\mathcal{P}}art}$

associated to $(G,\mathcal{H})$ and $(G^{\prime},\mathcal{H}^{\prime})$
respectively. Upon taking colimits, we obtain a map of partial groups from
${\mathbb{M}}(G,\mathcal{H})$ to
${\mathbb{M}}(G^{\prime},\mathcal{H}^{\prime})$, see Definition 3.6. This way
we obtain a functor,

(M) ${\mathbb{M}}(-,-)\colon{{\mathcal{D}}ecGraphs}\to{{\mathcal{P}}art},$

that we will investigate below.

##  4\. Graph morphisms from partial groups homomorphisms

To further study the functor (M), we consider the following categories.

######  Definition 4.1.

The category ${{\mathcal{F}}inDecGraphs}$ is the full subcategory of the
category ${{\mathcal{D}}ecGraphs}$ with objects the pairs $(G,\mathcal{H})$,
where $\mathcal{H}:=\\{H_{v}\\}_{v\in V}$ is a collection of _non-trivial
finite_ groups.

######  Definition 4.2.

The category ${{\mathcal{I}}njPart}$ is the wide subcategory of the category
${{\mathcal{P}}art}$ with morphisms the partial group homomorphisms whose
kernel has no torsion.

We show below that the functor (M) takes values in ${{\mathcal{I}}njPart}$ and
that the category ${{\mathcal{F}}inDecGraphs}$ embeds in the category
${{\mathcal{I}}njPart}$.

######  Theorem 4.3.

The functor

${\mathbb{M}}(-,-)\colon{{\mathcal{F}}inDecGraphs}\to{{\mathcal{I}}njPart},$

is full and faithful.

###### Proof.

First we check that for a morphism in ${{\mathcal{D}}ecGraphs}$,

$(f_{G},f_{\mathcal{H}})\colon(G,\mathcal{H})\to(G^{\prime},\mathcal{H}^{\prime}),$

the morphism ${\mathbb{M}}(f_{G},f_{\mathcal{H}})$ belongs to
${{\mathcal{I}}njPart}$. In fact, let
$x\in\ker({\mathbb{M}}(f_{G},f_{\mathcal{H}}))$ have finite order. Then, by
Lemma 3.7.(b), there exists $v\in V$ such that $x\in H_{v}$. If $x\neq 1$,
then $(x)\in\operatorname{CR}(\\{v\\})$ is cyclically reduced and
${\mathbb{M}}(f_{G},f_{\mathcal{H}})(x)=1=f_{v}(x)$. As $f_{v}$ is injective,
we get that $x=1$, a contradiction.

Now, for $(G,\mathcal{H})$ and $(G^{\prime},\mathcal{H}^{\prime})$ in
${{\mathcal{F}}inDecGraphs}$ and a morphism

$f\in\operatorname{Mor}_{{\mathcal{I}}njPart}({\mathbb{M}}(G,\mathcal{H}),{\mathbb{M}}(G^{\prime},\mathcal{H}^{\prime}))$

induced by the map of sets

$f\colon{\mathcal{M}}(G,\mathcal{H})\to{\mathcal{M}}(G^{\prime},\mathcal{H}^{\prime}),$

we will construct a morphism of decorated graphs

$(f_{G},f_{\mathcal{H}})\colon(G,\mathcal{H})\to(G^{\prime},\mathcal{H}^{\prime})$

such that ${\mathbb{M}}(f_{G},f_{\mathcal{H}})=f$. Fix a vertex $v\in V$.
Then, by Lemma 3.7.(a), $H_{v}$ is a non-trivial finite subgroup of
${\mathbb{M}}(G,\mathcal{H})$ and, as $\ker(f)$ has no torsion, the
restriction $f_{|H_{v}}$ is injective. Thus, by Lemma 2.7, $f(H_{v})$ is a
non-trivial finite subgroup of
${\mathbb{M}}(G^{\prime},\mathcal{H}^{\prime})$. By Lemma 3.7.(b), there
exists a unique $v^{\prime}\in V^{\prime}$ such that $f(H_{v})\subseteq
H^{\prime}_{v^{\prime}}$ and the group homomorphism

$f_{|H_{v}}\colon H_{v}\to H^{\prime}_{v^{\prime}}$

is injective. The correspondence $v\mapsto f_{G}(v):=v^{\prime}$ defines a map
$f_{G}\colon V\to V^{\prime}$ from the vertices of $G$ to those of
$G^{\prime}$. In fact, this map is a graph homomorphism: for every
$\\{v,w\\}\in E$ and every element $(h_{v},h_{w})\in H_{v}\times H_{w}$ with
$h_{v}\neq 1\neq h_{w}$, $((h_{v}),(h_{w}))\in\mathbb{D}(G,\mathcal{H})$ by
Lemma 3.7.(c). Thus, if we apply $f$, we get that
$((f(h_{v})),(f(h_{w})))\in\mathbb{D}(G^{\prime},\mathcal{H}^{\prime})$ and,
by Lemma 3.7.(c) again, $\\{f_{V}(v),f_{V}(w)\\}\in E$.

So me may define the morphism
$(f_{G},f_{\mathcal{H}})\colon(G,\mathcal{H})\to(G^{\prime},\mathcal{H}^{\prime})$
in ${{\mathcal{F}}inDecGraphs}$, where $f_{G}$ is the graph homomorphism
constructed above and $f_{\mathcal{H}}=\\{f_{v}\\}_{v\in V}$ with

(1) $f_{v}=f_{|H_{v}}$

for each vertex $v$ of $G$. Let
$u=(g_{1},\ldots,g_{r})\in\operatorname{CR}(C)$ for a clique
$C=\\{v_{1},\ldots,v_{n}\\}$ of $G$ and with with $g_{j}\in H_{v_{i_{j}}}$,
see Equation (S). Note that $(g_{i})\in\operatorname{CR}(C)$ for all $i$ and
that, as $\Pi(G,\mathcal{H})$ is given by reduced form,
$\Pi(G,\mathcal{H})((g_{1}),\ldots,(g_{r}))=(g_{1},\ldots,g_{r})$. Then we
also have that,

$\displaystyle f(g_{1},\ldots,g_{r})$ $\displaystyle=$ $\displaystyle
f(\Pi(G,\mathcal{H})((g_{1}),\ldots,(g_{r})))$
$\displaystyle\stackrel{{\scriptstyle\textrm{(H2)}}}{{=}}$
$\displaystyle\Pi(G^{\prime},\mathcal{H}^{\prime})(\overline{f}((g_{1}),\ldots,(g_{r})))$
$\displaystyle=$
$\displaystyle\Pi(G^{\prime},\mathcal{H}^{\prime})((f(g_{1})),\ldots,(f(g_{r})))$
$\displaystyle=$
$\displaystyle\Pi(G^{\prime},\mathcal{H}^{\prime})((f_{|H_{v_{i_{1}}}}(g_{1})),\ldots,(f_{|H_{v_{i_{r}}}}(g_{r})))$
$\displaystyle\stackrel{{\scriptstyle\eqref{equ:deffv}}}{{=}}$
$\displaystyle\Pi(G^{\prime},\mathcal{H}^{\prime})((f_{v_{i_{1}}}(g_{1})),\ldots,(f_{v_{i_{r}}}(g_{r})))$
$\displaystyle\stackrel{{\scriptstyle\eqref{equ:varphi}}}{{=}}$
$\displaystyle\Pi(G^{\prime},\mathcal{H}^{\prime})(\overline{\varphi_{(f_{G},f_{\mathcal{H}})}}((g_{1}),\ldots,(g_{r})))$
$\displaystyle\stackrel{{\scriptstyle\textrm{(H2)}}}{{=}}$
$\displaystyle\varphi_{(f_{G},f_{\mathcal{H}})}(\Pi(G,\mathcal{H})((g_{1}),\ldots,(g_{r})))$
$\displaystyle=$
$\displaystyle\varphi_{(f_{G},f_{\mathcal{H}})}(g_{1},\ldots,g_{r}).$

Thus $\varphi_{(f_{G},f_{\mathcal{H}})}=f$,
${\mathbb{M}}(f_{G},f_{\mathcal{H}})=f$, and the functor ${\mathbb{M}}(-,-)$
is full. It remains to prove that this functor is faithful but this is
straightforward: if we start with a morphism

$(f_{G},f_{\mathcal{H}})\in\operatorname{Mor}_{{\mathcal{F}}inDecGraphs}((G,\mathcal{H}),(G,\mathcal{H}^{\prime}))$

and apply the earlier construction to

$f={\mathbb{M}}(f_{G},f_{\mathcal{H}})\in\operatorname{Mor}_{{\mathcal{I}}njPart}({\mathbb{M}}(G,\mathcal{H}),{\mathbb{M}}(G^{\prime},\mathcal{H}^{\prime})),$

it is immediate that we recover the morphism $(f_{G},f_{\mathcal{H}})$. ∎

If we particularise the last result to automorphisms, we get the following
exact sequence.

######  Theorem 4.4.

Let $(G,\mathcal{H})\in{{\mathcal{F}}inDecGraphs}$. Then there exists an exact
sequence

(2) $1\to\prod_{v\in
V}\operatorname{Aut}_{{\mathcal{G}}roups}(H_{v})\to\operatorname{Aut}_{{\mathcal{P}}art}\big{(}{\mathbb{M}}(G,\mathcal{H})\big{)}\to\operatorname{Aut}_{{\mathcal{G}}raphs}(G).$

###### Proof.

Because automorphisms have trivial kernel, we have that

$\operatorname{Aut}_{{\mathcal{P}}art}\big{(}{\mathbb{M}}(G,\mathcal{H})\big{)}=\operatorname{Aut}_{{\mathcal{I}}njPart}\big{(}{\mathbb{M}}(G,\mathcal{H})\big{)}.$

Then, by Theorem 4.3, we have a bijection

$\textstyle{\operatorname{Aut}_{{\mathcal{F}}inDecGraphs}(G,\mathcal{H})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Aut}_{{\mathcal{I}}njPart}\big{(}{\mathbb{M}}(G,\mathcal{H})\big{)}}$$\textstyle{(f_{G},f_{\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{M}}(f_{G},f_{\mathcal{H}}).}$

We define the map
$\operatorname{Aut}_{{\mathcal{P}}art}\big{(}{\mathbb{M}}(G,\mathcal{H})\big{)}\to\operatorname{Aut}_{{\mathcal{G}}raphs}(G)$
in the statement by sending an automorphism of the partial group
${\mathbb{M}}(G,\mathcal{H})$ to its corresponding graph automorphism $f_{G}$.
If $f_{G}=\operatorname{id}_{G}$ then, for all $v\in V$, $f|_{H_{v}}$ is an
automorphism of $H_{v}$. Finally, any collection $(f_{v})_{v\in
V}\in\prod_{v\in V}\operatorname{Aut}(H_{v})$ induces via the bijection above
the partial group automorphism,

${\mathbb{M}}(id_{G},\\{f_{v}\\}_{v\in
V})\colon{\mathbb{M}}(G,\mathcal{H})\to{\mathbb{M}}(G,\mathcal{H}),$

and the corresponding map $\prod_{v\in
V}\operatorname{Aut}(H_{v})\to\operatorname{Aut}_{{\mathcal{P}}art}\big{(}{\mathbb{M}}(G,\mathcal{H})\big{)}$
is injective. ∎

######  Remark 4.5.

Observe that although the sequence (2) above is exact, it may not be short
exact. For example, let $K_{2}$ be the complete graph on two vertices, labeled
$a$ and $b$, and let $\mathcal{H}$ be the collection of finite groups given by
$H_{a}={\mathbb{Z}}_{2}$ and $H_{b}={\mathbb{Z}}_{3}$. Then
$\operatorname{Aut}_{{\mathcal{G}}roups}(H_{a})=\\{1\\}$,
$\operatorname{Aut}_{{\mathcal{G}}roups}(H_{b})={\mathbb{Z}}_{2}$ and
$\operatorname{Aut}_{{\mathcal{G}}raphs}(K_{2})={\mathbb{Z}}_{2}$. The cliques
in $K_{2}$ are $\Delta=\\{\\{a\\},\\{b\\},\\{a,b\\}\\}$ and we have

$\displaystyle\operatorname{CR}(\\{a\\})=$ $\displaystyle\\{(a)\\}\text{,
}\operatorname{CR}(\\{b\\})=\\{(b),(b^{2})\\}\text{, and }$
$\displaystyle\operatorname{CR}(\\{a,b\\})=$
$\displaystyle\\{(a),(b),(b^{2})\\}\cup\\{(a,b^{\epsilon_{1}},\ldots,a,b^{\epsilon_{n}}),n\geq
0,\epsilon_{i}\in\\{1,2\\}\\}\cup$
$\displaystyle\\{(b^{\epsilon_{1}},a,\ldots,b^{\epsilon_{n}},a),n\geq
0,\epsilon_{i}\in\\{1,2\\}\\}.$

where we have identified the vertices with the element of order 2 or 3 in the
associated decorating group. From here, an easy calculation shows that the
only non-trivial automorphism of ${\mathbb{M}}(G,\mathcal{H})$ is given by
$a\mapsto a,b\mapsto b^{2},b^{2}\mapsto b$. Hence
$\operatorname{Aut}_{{\mathcal{P}}art}\big{(}{\mathbb{M}}(G,\mathcal{H})\big{)}={\mathbb{Z}}_{2}$
and the sequence (2) becomes

$1\to{\mathbb{Z}}_{2}\to{\mathbb{Z}}_{2}\to{\mathbb{Z}}_{2},$

which cannot be exact.

######  Remark 4.6.

A particular case for which the sequence (2) is short exact is given by the
condition

$H_{v}=H_{v^{\prime}}\text{ for all $v,v^{\prime}\in V$.}$

Under this condition, assuming the hypotheses in Theorem 4.4, and writing $A$
for the common group $H_{v}$ for any $v\in V$, the map

$\textstyle{\operatorname{Aut}_{{\mathcal{G}}raphs}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Aut}_{{\mathcal{P}}art}\big{(}{\mathbb{M}}(G,\mathcal{H})\big{)}\overset{\cong}{\to}\operatorname{Aut}_{{\mathcal{F}}inDecGraphs}(G,\mathcal{H})}$$\textstyle{f_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(f_{G},\\{\operatorname{id}_{A}\colon
A_{v}\to A_{f_{G}(v)}\\}_{v\in V})}$

is a section of the map
$\operatorname{Aut}_{{\mathcal{P}}art}\big{(}{\mathbb{M}}(G,\mathcal{H})\big{)}\to\operatorname{Aut}_{{\mathcal{G}}raphs}(G)$
in (2). Hence the sequence is short exact. In addition, we have the wreath
product decomposition

$\displaystyle\operatorname{Aut}_{{\mathcal{P}}art}\big{(}{\mathbb{M}}(G,\mathcal{H})\big{)}$
$\displaystyle\cong\overbrace{\operatorname{Aut}(A)\times\ldots\times\operatorname{Aut}(A)}^{|V|\text{
times}}\rtimes\operatorname{Aut}_{{\mathcal{G}}raphs}(G)$
$\displaystyle\cong\operatorname{Aut}(A)\wr\operatorname{Aut}_{{\mathcal{G}}raphs}(G),$

where we have used that every injective endomorphism of a finite group is an
automorphism.

##  5\. Path partial groups

In this section, we discuss how the algebraic structure of the partial group
associated to a decorated undirected graph (Definition 3.6) is related to path
concatenation within $G$ in a way that resembles the path algebra
corresponding to a directed graph. The main difference is that in the path
algebra two non-composable paths multiply to $0$, while in our construction
such a product does not exist. We first consider the following functor,

$\textstyle{{{\mathcal{G}}raphs}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}inDecGraphs}$$\textstyle{G=(V,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(G,\\{{\mathbb{Z}}_{2}\\}_{v\in
V}).}$

So a graph is sent to itself decorated with the group ${\mathbb{Z}}_{2}$ on
each vertex. On morphisms, the graph homomorphism $f_{G}\colon G\to
G^{\prime}$ is sent to the morphism

$(f_{G},\\{\operatorname{id}_{{\mathbb{Z}}_{2}}\colon({\mathbb{Z}}_{2})_{v}\to({\mathbb{Z}}_{2})_{f_{G}(v)}\\}_{v\in
V}).$

Below we prove that the category ${{\mathcal{G}}raphs}$ embeds into the
category ${{\mathcal{I}}njPart}$.

######  Lemma 5.1.

The composition functor

$\textstyle{{{\mathcal{G}}raphs}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\mathcal{F}}inDecGraphs}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathbb{M}}(-,-)}$$\textstyle{{{\mathcal{I}}njPart},}$

is full and faithful.

###### Proof.

Because of Theorem 4.3, it is enough to prove that for any two graphs $G$ and
$G^{\prime}$ in ${{\mathcal{G}}raphs}$, the map

$\textstyle{\operatorname{Mor}_{{\mathcal{G}}raphs}(G,G^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Mor}_{{\mathcal{F}}inDecGraphs}((G,\\{{\mathbb{Z}}_{2}\\}_{v\in
V}),(G^{\prime},\\{{\mathbb{Z}}_{2}\\}_{v\in
V}))}$$\textstyle{f_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(f_{G},\\{\operatorname{id}_{{\mathbb{Z}}_{2}}\colon({\mathbb{Z}}_{2})_{v}\to({\mathbb{Z}}_{2})_{f_{G}(v)}\\}_{v\in
V})}$

is a bijection, and this is straightforward as the only injective group
homomorphism ${\mathbb{Z}}_{2}\to{\mathbb{Z}}_{2}$ is the identity. ∎

In the next three paragraphs, we provide an informal explanation of how the
construction of path algebras inspired the construction of the path partial
group

${\mathbb{P}}(G)={\mathbb{M}}(G,\\{{\mathbb{Z}}_{2}\\}_{v\in V})$

for an undirected graph $G$. To justify this notation recall that, by Equation
(S), the elements in the alphabet
${\mathcal{M}}={\mathcal{M}}(G,\\{{\mathbb{Z}}_{2}\\}_{v\in V})$ are the
cyclically reduced words $u=(g_{1},g_{2},\ldots,g_{r})\in\operatorname{CR}(C)$
for some clique $C=\\{v_{1},\ldots,v_{n}\\}$ of $G$, with $g_{i}\in v_{i_{j}}$
for $i=1,\ldots,r$. As the elements $g_{i}$ must be non-trivial elements of
${\mathbb{Z}}_{2}$, the cyclically reduced word $u$ is completely determined
by the sequence of vertices

(3) $(v_{i_{1}},v_{i_{2}},\ldots,v_{i_{r}}).$

As all these vertices belong to the clique $C$, this sequence can be thought
of as the path
$(\\{{v_{i_{1}}},v_{i_{2}}\\},\ldots,\\{v_{i_{r-1}},v_{i_{r}}\\})$ in $G$. For
$r=1$, we have the path consisting of a single vertex, $(v_{i_{1}})$. Thus
elements in ${\mathcal{M}}$ can be seen as _paths_ in the graph $G$ with $1$
or more vertices.

By Definition 2.1.(P3), there are inverses on every partial group. In
particular, the inverse of a vertex should be itself, and this explains the
choice of ${\mathbb{Z}}_{2}$ in the embedding
${{\mathcal{G}}raphs}\to{{\mathcal{F}}inDecGraphs}$. Moreover, the inverse of
a path should be the reversed path and thus, if $G$ were directed, it should
contain the reversed arrow of each arrow. This is why we ask $G$ to be an
undirected graph. As described above, we only consider paths in $G$ that are
contained in some clique of $G$. The reason for this is that the existence of
inverses imply cancellations. For instance, assume that we have two paths in
$G$ given by $(a,b)$ and $(b,c)$, where $a,b,c$ are vertices of $G$. If we
multiply them, we would obtain

$\Pi((a,b)\circ(b,c))=\Pi(a,b,b,c)=(a,c),$

where the two consecutive elements $b$ cancel each other as they correspond to
the same element in ${\mathbb{Z}}_{2}$. Then $(a,c)$ should be a path in $G$
and the edge $\\{a,c\\}$ should be contained in $G$. Hence, paths that may be
multiplied should have vertices contained in some clique of $G$.

Finally, we wanted ${\mathbb{P}}(G)$ to have partial group automorphisms
“close” to the graphs automorphisms of $G$. In particular, the inner
automorphism given by conjugation by any vertex should not produce an
automorphism of ${\mathbb{P}}(G)$, since inner automorphisms are what prevents
the category ${{\mathcal{G}}roups}$ to be universal [20]. This explains why we
only consider paths in $G$ which are not closed or, equivalently, words in
${\mathcal{M}}$ that are cyclically reduced.

That the constraints we have imposed are correct is supported by the following
results.

######  Theorem 5.2.

Let $G\in Ob({{\mathcal{G}}raphs})$. Then

$\operatorname{Aut}_{{\mathcal{P}}art}\big{(}{\mathbb{P}}(G))\cong\operatorname{Aut}_{{\mathcal{G}}raphs}(G).$

###### Proof.

Since $\operatorname{Aut}({\mathbb{Z}}_{2})$ is trivial, the exact sequence
(2) in Theorem 4.4 gives rise to an injective homomorphism
$\Psi\colon\operatorname{Aut}_{{\mathcal{P}}art}({\mathbb{P}}(G))\to\operatorname{Aut}_{{\mathcal{G}}raphs}(G)$.
In addition, the composition functor in Lemma 5.1 induces a group homomorphism

$\Phi\colon\operatorname{Aut}_{{\mathcal{G}}raphs}(G)\to\operatorname{Aut}_{{\mathcal{P}}art}({\mathbb{P}}(G)),$

which is nothing but the section of (2) constructed in Remark 4.6, hence
$\Psi\circ\Phi=\operatorname{id}_{\operatorname{Aut}_{{\mathcal{G}}raphs}(G)}$.
Therefore, they are both isomorphisms inverse to each other. ∎

######  Theorem 5.3.

The category ${{\mathcal{P}}art}$ is universal. Moreover, given an abstract
group $H$ there exist infinitely many non isomorphic partial groups
${\mathbb{M}}$ such that
$\operatorname{Aut}_{{\mathcal{P}}art}({\mathbb{M}})\cong H$.

###### Proof.

Let $H$ be an abstract group. According to [22, 19, 27], there exists
infinitely many non isomorphic simple graphs $G=(V,E)$, such that
$\operatorname{Aut}_{{\mathcal{G}}raphs}(G)\cong H$. Then, these non
isomorphic graphs $G$ give rise to partial groups ${\mathbb{P}}(G)$ which are
not isomorphic by Lemma 5.1, and such that
$\operatorname{Aut}_{{\mathcal{P}}art}({\mathbb{P}}(G))\cong H$ by Theorem
5.2. ∎

A rigid object in a category $\mathcal{C}$ is an object $C$ such that
$\operatorname{Aut}_{\mathcal{C}}(C)=\\{1\\}$. Observe that, while there
exists just one non-trivial rigid object in ${{\mathcal{G}}roups}$, i.e.,
${\mathbb{Z}}_{2}$ is the only non-trivial group $K$ such that
$\operatorname{Aut}_{{\mathcal{G}}roups}(K)=\\{1\\}$, the situation in
${{\mathcal{P}}art}$ is drastically different: Theorem 5.3 ensures that there
exist infinitely many non-trivial rigid partial groups. In the category of
topological spaces, there exist also many non-trivial topological spaces whose
group of self-homotopy equivalences is $\\{1\\}$. For instance, for the
Eilenberg-MacLane spaces $K({\mathbb{Z}}_{2},n)$, $n\geq 0$, we have

$[K({\mathbb{Z}}_{2},n),K({\mathbb{Z}}_{2},n)]=H^{n}(K({\mathbb{Z}}_{2},n);{\mathbb{Z}}_{2})=\operatorname{Hom}({\mathbb{Z}}_{2},{\mathbb{Z}}_{2})={\mathbb{Z}}_{2},$

and so the identity is the unique self-homotopy equivalence of
$K({\mathbb{Z}}_{2},n)$. Other rigid non-trivial topological spaces not having
the homotopy type of an Eilenberg-MacLane space can be found in [16, 14].
Thus, regarding rigidity, the category of partial groups behaves much more
like topological spaces and not like groups. This is not surprising as every
partial group can be naturally seen as a simplicial set. In the next section
we employ this fact to give a solution to the Kahn realizability problem.

##  6\. Kahn realizability problem

The Kahn realizability problem for groups and topological spaces consists of,
given a group $G$, find a topological space $X$ whose group of homotopy
classes of pointed self-homotopy equivalences is isomorphic to $G$. If $G$ is
finite, in [16], C. Costoya and the third author of this work show that there
exist infinitely many non-homotopy equivalent rational elliptic spaces $X$
that realize $G$ in that sense. Benkhalifa asserts in [6] that the finiteness
assumption on $G$ may be dropped if one considers a
${\mathbb{Z}}_{(p)}$-localized space $X$ instead. These two works construct
spaces in the category of $CW$-complexes. Below we use the tools developed so
far to construct a simplicial set whose group of homotopy classes of pointed
simplicial self-homotopy equivalences is a given prescribed arbitrary group.

As mentioned at the end of the previous section, every partial group is
naturally a simplicial set, see [23, Theorem 4.8] and [7, Definition 2.1]. In
more detail, if we have a partial group
${\mathbb{M}}=\left({\mathcal{M}},\mathbb{D},\Pi,(-)^{-1}\right)$, we
naturally think of it as a simplicial set by declaring $n$-simplices to be
words of length $n$ in $\mathbb{D}$ and defining the face and degeneracy
operators via the product and insertion of $1$ respectively. This way, one
obtains a reduced simplicial set, i.e., a simplicial set with a single vertex
which, in this case, corresponds to the empty word $\emptyset$.

If ${\mathcal{K}}$ is a simplicial set, we denote by
${\mathcal{E}}({\mathcal{K}})$ the group of homotopy classes of pointed
simplicial self-homotopy equivalences of ${\mathcal{K}}$ whenever it is well
defined. In general, the set of homotopy classes of simplicial maps between
two simplicial sets can not be defined as the homotopy relation is not
necessarily transitive. A sufficient condition is that the target simplicial
set is a Kan complex [25, Definition 1.3], which is not the case for a partial
group unless it is a group, see [7, Remark 2.3]. Nevertheless, in any partial
group ${\mathbb{M}}$, inner horns can be “filled”, i.e., ${\mathbb{M}}$ is a
quasicategory, also known as an $\infty$-category. This implies that the group
of homotopy classes of pointed simplicial self-homotopy equivalences
${\mathcal{E}}({\mathbb{M}})$ do exist, see [26, 11.11,20.4]. Furthermore, as
proven in [7] and explained in the proof below, the group
${\mathcal{E}}({\mathbb{M}})$ for any partial group ${\mathbb{M}}$ can be
obtained as a quotient of the group
$\operatorname{Aut}_{{\mathcal{P}}art}({\mathbb{M}})$.

######  Theorem 6.1.

Given an abstract group $H$, there exist infinitely many non-homotopy
equivalent simplicial sets ${\mathcal{K}}$ such that
${\mathcal{E}}({\mathcal{K}})\cong H$.

###### Proof.

Let
${\mathbb{M}}={\mathbb{P}}(G)=\left({\mathcal{M}},\mathbb{D},\Pi,(-)^{-1}\right)$
be one of the infinitely many non isomorphic partial groups ${\mathbb{M}}$
such that $\operatorname{Aut}_{{\mathcal{P}}art}({\mathbb{M}})\cong H$
provided by Theorem 5.3, where $G$ is some simple graph. First note that, as
${\mathbb{M}}$ is a reduced simplicial set, every self-homotopy equivalence of
${\mathbb{M}}$ is pointed. Now, by [7, Lemma 2.12], every homotopy equivalence
${\mathbb{M}}\to{\mathbb{M}}$ is an automorphism and, by [7, Proposition
2.11], the homotopy relation in
$\operatorname{Aut}_{{\mathcal{P}}art}({\mathbb{M}})$ is an equivalence
relation and we have an exact sequence

$N({\mathbb{M}})\to\operatorname{Aut}_{{\mathcal{P}}art}({\mathbb{M}})\to{\mathcal{E}}({\mathbb{M}})\to
1.$

Here, the _normalizer_ $N({\mathbb{M}})$ is a subgroup of ${\mathbb{M}}$ whose
image in $\operatorname{Aut}_{{\mathcal{P}}art}({\mathbb{M}})$ is a normal
subgroup there, see [7, Definition 2.8]. By Equation (3), we may identify the
elements of ${\mathcal{M}}$ with the sequences of vertices
$(v_{i_{1}},v_{i_{2}},\ldots,v_{i_{r}})$ such that these vertices belong to a
clique $C$ of the graph $G$ and such that (cyclically) consecutive vertices
are different. Then, by [7, Lemma 2.9(b)], if
$\eta=(v_{i_{1}},v_{i_{2}},\ldots,v_{i_{r}})$ belongs to $N({\mathbb{M}})$ and
$\mu=(v^{\prime}_{i_{1}},v^{\prime}_{i_{2}},\ldots,v^{\prime}_{i_{s}})$ is an
arbitrary element of ${\mathcal{M}}$, we must have that the word
$(\eta,\mu,\eta^{-1})$ belongs to $\mathbb{D}$. If $r\geq 1$, we can take
$\mu=(v_{i_{1}})$ to obtain, by Equation (H), that

$(v_{i_{1}},v_{i_{2}},\ldots,v_{i_{r}},v_{i_{1}},v_{i_{r}},\ldots,v_{i_{1}})$

belongs to $\operatorname{CR}(C)$, but this cannot be the case as the first
and last vertices of this sequence are equal. So $r=0$, $\eta$ is the empty
word and $N({\mathbb{M}})=1$. Then, from the exact sequence above we obtain
that
${\mathcal{E}}({\mathbb{M}})\cong\operatorname{Aut}_{{\mathcal{P}}art}({\mathbb{M}})\cong
H$.

If ${\mathbb{M}}^{\prime}$ is another partial group with
$\operatorname{Aut}_{{\mathcal{P}}art}({\mathbb{M}})\cong H$ provided by
Theorem 5.3 and which is homotopy equivalent to ${\mathbb{M}}$ then, by [7,
Lemma 2.12] again, we have that ${\mathbb{M}}$ and ${\mathbb{M}}^{\prime}$ are
isomorphic. This finishes the proof. ∎

##  7\. The graph of maximal locally finite subgroups of a partial group

A group $H$ is called _locally finite_ if every finitely generated subgroup of
$H$ is finite. In this section we construct a graph out of the maximal locally
finite subgroups of a given partial group. Recall that a subgroup of a partial
group is a partial subgroup which is itself a group, see Definition 2.5. This
will allow us to show that, given a simple undirected graph $G=(V,E)$ and a
collection of non-trivial locally finite groups
$\mathcal{H}:=\\{H_{v}\\}_{v\in V}$, $G$ can be fully recovered (up to
isomorphism) from the algebraic structure of ${\mathbb{M}}(G,\mathcal{H})$.

We first prove that in a partial group there is at least a maximal locally
finite subgroup.

######  Lemma 7.1.

Let ${\mathbb{M}}=\left({\mathcal{M}},\mathbb{D},\Pi,(-)^{-1}\right)$ be a
partial group. Then the poset of the locally finite subgroups of
${\mathbb{M}}$ ordered by inclusion has a maximal element.

###### Proof.

Let ${\mathcal{X}}$ be the set of locally finite subgroups of $G$. Notice that
${\mathcal{X}}$ is not empty since $\\{1\\}$ is a locally finite subgroup of
${\mathbb{M}}$. If $H_{1}\leq H_{2}\leq\cdots$ is an increasing chain of
locally finite subgroups of ${\mathbb{M}}$, consider the subgroup
$H_{\infty}=\bigcup_{i\in{\mathbb{N}}}H_{i}\leq H$. By construction, every
finitely generated subgroup of $H_{\infty}$ lies in some $H_{i}$ for $i$ big
enough. Therefore, $H_{\infty}$ is a locally finite subgroup of $H$ and thus
an upper bound for the given chain of elements of ${\mathcal{X}}$. Applying
Zorn’s Lemma, we obtain that ${\mathcal{X}}$ has a maximal element. ∎

######  Definition 7.2.

Let ${\mathbb{M}}=\left({\mathcal{M}},\mathbb{D},\Pi,(-)^{-1}\right)$. The
_maximal locally finite subgroup graph_ of ${\mathbb{M}}$ is the graph
$\operatorname{MaxSub_{loc}}({\mathbb{M}})$ with vertices the set of non-
trivial maximal locally finite subgroups of ${\mathbb{M}}$ and declaring two
_distinct_ vertices $H_{1}$ and $H_{2}$ to be adjacent if and only if there
exist $h_{1}\in H_{1}^{*}$ and $h_{2}\in H_{2}^{*}$ such that
$(h_{1},h_{2})\in\mathbb{D}$.

######  Remark 7.3.

Notice that $\operatorname{MaxSub_{loc}}({\mathbb{M}})$ is a well-defined
simple undirected graph. Indeed, given two different non-trivial maximal
locally finite subgroups $H_{1}$ and $H_{2}$ of ${\mathbb{M}}$ and two
elements $h_{1}\in H_{1}^{*}$ and $h_{2}\in H_{2}^{*}$, we have

$(h_{1},h_{2})\in\mathbb{D}\text{ if and only if
}(h_{2}^{-1},h_{1}^{-1})\in\mathbb{D}.$

Thus, the adjacency relation considered is symmetric.

Notice that the graph $\operatorname{MaxSub_{loc}}({\mathbb{M}})$ may be empty
if ${\mathbb{M}}$ has no non-trivial subgroups, and Example 2.4 provides an
example of such situation. In Definition 7.2, we look at locally finite
subgroups instead of at subgroups because in a path partial group every edge
give rise to a copy of ${\mathbb{Z}}$. This is demonstrated in Example 7.4
below.

######  Example 7.4.

Let $K_{2}$ be the complete graph on two vertices, labeled $a$ and $b$. Then
the Hasse diagram of the poset of cliques $\Delta$ of $K_{2}$ is the
following,

$\textstyle{\\{a,b\\}}$$\textstyle{\\{a\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{b\\}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Recall that
${\mathbb{P}}(K_{2})=\left({\mathcal{M}},\mathbb{D},\Pi,(-)^{-1}\right)$ is
the colimit in the category ${{\mathcal{P}}art}$ of the partial groups

${\mathbb{M}}(C)=(\operatorname{CR}(C),\mathbb{D}(C),\Pi(C),(-)_{C}^{-1})$

for $C\in\Delta$. Notice that

$\displaystyle\operatorname{CR}(\\{a\\})=\\{(a)\\}\text{,
}\operatorname{CR}(\\{b\\})=\\{(b)\\}\text{, and }$
$\displaystyle\operatorname{CR}(\\{a,b\\})=\\{(a),(b)\\}\cup\\{\underbrace{(a,b,\ldots,a,b)}_{2n},n\geq
0\\}\cup\\{\underbrace{(b,a,\ldots,b,a)}_{2n},n\geq 0\\},$

where we have identified the vertices with the element of order 2 in the
associated decorating group. The maximal subgroups in ${\mathbb{P}}(K_{2})$
are $\langle(a)\rangle\cong{\mathbb{Z}}_{2}$,
$\langle(b)\rangle\cong{\mathbb{Z}}_{2}$, and
$\langle(a,b)\rangle\cong{\mathbb{Z}}$. Only two of these three subgroups are
locally finite and it is straightforward that
$\operatorname{MaxSub_{loc}}({\mathbb{P}}(K_{2}))$ is the complete graph on
two vertices.

There are alternative definitions for the _maximal locally finite subgroup
graph_ associated to a partial group that make sense but that we do not employ
here. For instance, one could consider the same set of vertices as that in
Definition 7.2 but endowed with the “stronger” notion of adjacency described
as follows: $H_{1}$ and $H_{2}$ are adjacent if and only if _for all_
$h_{1}\in H_{1}^{*}$ and _for all_ $h_{2}\in H_{2}^{*}$,
$(h_{1},h_{2})\in\mathbb{D}$. In the particular case of path partial groups
both definitions agree, but in general non-isomorphic graphs are obtained as
the next example shows.

######  Example 7.5.

Consider the dihedral group of size $8$,

$D_{8}=\langle x,t\;|\;x^{4}=1,t^{2}=1,txt=x\rangle,$

its center, and its two Klein four-groups,

$Z=Z(D_{8})=\langle x^{2}\rangle\text{, }V=\langle x^{2},t\rangle\text{ and
}V^{\prime}=\langle x^{2},tx\rangle.$

Define the partial group
${\mathbb{M}}=\left({\mathcal{M}},\mathbb{D},\Pi,(-)^{-1}\right)$ as the
colimit in the category ${{\mathcal{P}}art}$ (see Section 2) of the diagram
given by inclusions among these three subgroups,

${\mathbb{M}}=\mathop{colim}_{{{\mathcal{P}}art}}(V\leftarrow Z\rightarrow
V^{\prime}).$

Then the set of maximal locally finite subgroups of ${\mathbb{M}}$ is exactly
$\\{V,V^{\prime}\\}$. Moreover, the word $(x^{2},x^{2})$ belongs to
$\mathbb{D}$ and hence the graph $\operatorname{MaxSub_{loc}}({\mathbb{M}})$
is connected. Nevertheless, the word $(t,tx)$ does not belong to $\mathbb{D}$
and hence the graph defined using the “stronger” condition in the paragraph
above is not connected. In particular, the partial group ${\mathbb{M}}$ cannot
be isomorphic to a partial group of the form ${\mathbb{M}}(G,\mathcal{H})$ for
any pair $(G,\mathcal{H})\in{{\mathcal{D}}ecGraphs}$.

######  Lemma 7.6.

Let $G=(V,E)$ be a simple undirected graph and $\mathcal{H}:=\\{H_{v}\\}_{v\in
V}$ be a collection of non-trivial locally finite groups. Then, for every non-
trivial locally finite subgroup $H$ of ${\mathbb{M}}(G,\mathcal{H})$, there
exist a unique $v\in V$ such that $H\leq H_{v}$. In particular, the set of
maximal locally finite subgroups of ${\mathbb{M}}(G,\mathcal{H})$ is
$\mathcal{H}$.

###### Proof.

Let $H\leq{\mathbb{M}}(G,\mathcal{H})$ be a non-trivial locally finite
subgroup. Then $H$ contains at least a non-trivial finite group $K$ and, by
Lemma 3.7.(b), there exists a vertex $v\in V$ such that $K\leq H_{v}$. Hence
$1\neq K\leq H_{v}\cap H$. Now assume that there are vertices $v_{1}$ and
$v_{2}$ in $V$ such that $H_{v_{1}}\cap H$ and $H_{v_{2}}\cap H$ are non-
trivial. Then we may choose $h_{1}\in(H_{v_{1}}\cap H)^{*}$ and
$h_{2}\in(H_{v_{2}}\cap H)^{*}$ and consider the finite subgroup $\langle
h_{1},h_{2}\rangle\leq H$ of ${\mathbb{M}}(G,\mathcal{H})$. Then, by Lemma
3.7.(b) again, there exists a unique vertex $v\in V$ such that $\langle
h_{1},h_{2}\rangle\leq H_{v}$. Since $\langle h_{1}\rangle\leq H_{v_{1}}$ is a
non-trivial finite subgroup which is contained in both $H_{v_{1}}$ and
$H_{v}$, the uniqueness in Lemma 3.7.(b) gives that $v_{1}=v$. The analogous
argument applied to $h_{2}$ shows that $v_{2}=v$ and hence $v_{1}=v_{2}$. This
proves the first part of the lemma.

For the second part, notice that for every $v\in V$, $H_{v}$ is a non-trivial
locally finite subgroup of ${\mathbb{M}}(G,\mathcal{H})$ by Lemma 3.7.(a).
Therefore, by what we just proved in the paragraph above, the $H_{v}$’s are
maximal locally finite subgroups of ${\mathbb{M}}(G,\mathcal{H})$ and every
maximal locally finite subgroup of ${\mathbb{M}}(G,\mathcal{H})$ is of this
form ∎

######  Theorem 7.7.

Let $G=(V,E)$ be a simple undirected graph and $\mathcal{H}:=\\{H_{v}\\}_{v\in
V}$ be a collection of non-trivial locally finite groups. Then
$\operatorname{MaxSub_{loc}}\big{(}{\mathbb{M}}(G,\mathcal{H})\big{)}\cong G$.

###### Proof.

By Lemma 7.6, the maximal locally finite subgroups of
${\mathbb{M}}(G,\mathcal{H})$ are exactly the elements of $\mathcal{H}$.
Moreover, thanks to Lemma 3.7.(c), $H_{v}$ and $H_{v^{\prime}}$ are adjacent
in $\operatorname{MaxSub_{loc}}\big{(}{\mathbb{M}}(G,\mathcal{H})\big{)}$ if
and only $\\{v,v^{\prime}\\}\in E$. Hence the application which sends a vertex
$v\in V$ to $H_{v}$ defines an isomorphism of graphs between the graph $G$ and
the graph
$\operatorname{MaxSub_{loc}}\big{(}{\mathbb{M}}(G,\mathcal{H})\big{)}$. ∎

Notice that by also considering the maximal locally finite subgroups
themselves, one recovers all the data, i.e., the graph and the decorating
groups. Theorem 7.7 lays the foundations to start the study of Problem 1.3: as
the isomorphism type of ${\mathbb{M}}(G,\mathcal{H})$ in ${{\mathcal{P}}art}$
determines the isomorphism type of $G$ in ${{\mathcal{G}}raphs}$, the graph
theoretical properties of $G$ should be reflected on the algebraic properties
of the partial group ${\mathbb{M}}(G,\mathcal{H})$ and vice versa.

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