A note on dual prehomomorphisms from a group into the Margolis–Meakin expansion of a group

Abstract

We give a category-free order theoretic variant of a key result in Auinger and Szendrei (J Pure Appl Algebra 204(3):493–506, 2006) and illustrate how it might be used to compute whether a finite X-generated group H admits a canonical dual prehomomorphism into the Margolis–Meakin expansion M(G) of a finite X-generated group G. We show that for G the Klein four-group a suitable H must be of exponent 6 at least and recapture a result of Szakács.

1 Introduction

The following note considers canonical, i.e. generator preserving dual prehomomorphisms from an X-generated group H into the Margolis–Meakin expansion M(G) of an X-generated group G. It was shown by Auinger and Szendrei [1] that such mappings play an important role in constructing (finite) F-inverse covers for (finite) inverse monoids. We give a necessary and sufficient order theoretic condition for M(G) to admit a canonical dual prehomomorphism from an X-generated group H. It can be seen as a variant of the key statement Lemma 3.1 in [1] and might be applicable on a computer. The idea is to represent the elements of both M(G) and H as congruence classes of words in the free monoid with involution $(X\cup X^{-1})^*$. This enables us to handle the elements of H in relation to M(G) by systematically going through the words in $(X\cup X^{-1})^*$. We use the slightly different view of M(G), introduced in [2], to show how already known positive examples fit into the picture. Further, for G the Klein four-group, we prove that a suitable group H must be of exponent 6 at least and recapture a result of Szakács [6]. It should be noted that in our construction the groups H, we consider as possible candidates for admitting a canonical dual prehomorphism into M(G), may be arbitrarily X-generated extensions by G. This is in contrast to [1], where H is assumed to be an X-generated subgroup of a semidirect product of a relatively free group by G.

2 Preliminaries and notations

For all undefined notions and notations, the reader is referred to [3, 5]. Let X be a nonempty set and let G be an X-generated group with respect to an injection $\varepsilon _G:X \rightarrow G\setminus \{1_G\}$. Note that the mapping $\varepsilon _G$ can be uniquely extended to a homomorphism $\varphi _G:(X\cup X^{-1})^* \rightarrow G$, where $(X\cup X^{-1})^*$ is the free monoid with involution on X. For $w\in (X\cup X^{-1})^*$ we denote $w\varphi _G$ by ${\overline{w}}$. By the Cayley graph $\Gamma (G)$ with respect to $\varepsilon _G$, we mean the directed graph whose vertex set $V(\Gamma (G))$ is G and whose edge set $E(\Gamma (G))$ is $G\times X$, where for each $g\in G, x\in X, (g,x)$ denotes an edge with initial vertex g and terminal vertex $g{\overline{x}}$. Put

$$\begin{aligned} M(G)= & {} \{(\Gamma ,g) :\Gamma \text { is a finite connected subgraph of } \Gamma (G) \text { with at least one edge} \\&\quad \text { and }1_G,g\in V(\Gamma )\} \cup \{(\emptyset ,1_G)\}. \end{aligned}$$

There is a natural action of G on the semilattice of all subgraphs of $\Gamma (G)$ with operation the set theoretic union, defined as follows: Put $g\emptyset = \emptyset$, and for each nonempty subgraph $\Gamma$ of $\Gamma (G)$ and $g\in G$, let $g\Gamma$ be the subgraph of $\Gamma (G)$ with $V(g\Gamma ) = \{gh:h\in V(\Gamma )\}$ and $E(g\Gamma ) = \{(gh,x):(h,x)\in E(\Gamma )\}$. The graphs we consider do not have isolated vertices, whence they are solely determined by their edge sets, and we conveniently may regard them as (possibly empty) subsets of $X\times G$.

The following theorem was essentially proved in [4].

Theorem 2.1

[4] M(G) is an E-unitary inverse monoid with respect to the multiplication $(\Gamma ,g)(\Gamma ^\prime ,h) = (\Gamma \cup g\Gamma ^\prime ,gh)$ with identity element $(\emptyset ,1_G)$ and maximal group homomorphic image G. Further, M(G) is X-generated as inverse monoid via the injection $\varepsilon _{M(G)}:x \mapsto (\{(1_G,x)\},{\overline{x}})$.

We often represent the elements of M(G) by their corresponding images $\left\langle w \right\rangle$ in $(X\cup X^{-1})^*/\ker \varphi _{M(G)}$, where $\varphi _{M(G)}$ denotes the unique extension of $\varepsilon _{M(G)}$ to a homomorphism from $(X\cup X^{-1})^*$ onto M(G). Then obviously $\left\langle \emptyset \right\rangle$ corresponds to $(\emptyset ,1_G)$. Let $\emptyset \ne w = \prod \limits _{i=1}^{n} x_i^{\eta _i},\,\eta _i \in \{-1,1\}$, be a word in $(X\cup X^{-1})^*$. To w we associate a word $w^{\prime } =h_1 x_1 h_2 x_2 \cdots h_n x_n h_{n+1}$ in the free product $X^* *G$, where $X^*$ is the free monoid on X, by replacing each $x_i^{\eta _i}$ in w by $g_i x_i g_i$, where

$$\begin{aligned} g_i ={\left\{ \begin{array}{ll} 1_G &{} \text {if }\; \eta _i = 1, \\ {\overline{x_i}}^{\,-1} &{} \text {if }\; \eta _i = -1. \end{array}\right. } \end{aligned}$$

Then $\left\langle w \right\rangle$ corresponds to $(\Gamma (\left\langle w \right\rangle ),{\overline{w}})\in M(G)$, in symbols $\left\langle w\right\rangle \widehat{=} (\Gamma (\left\langle w \right\rangle ),{\overline{w}})$, where $E(\Gamma (\left\langle w \right\rangle )) = \{(h_1,x_1),(h_1{\overline{x_1}}h_2,x_2), \dots , (h_1 {\overline{x_1}}h_2{\overline{x_2}}\cdots h_n,x_n)\}$ and ${\overline{w}} = \prod \limits _{i=1}^{n} {\overline{x_i}}^{\eta _i}$. Conversely, for each $(\Gamma , g)\in M(G)$ there is a unique $\left\langle w \right\rangle$ with $\left\langle w\right\rangle \widehat{=} (\Gamma ,g)$ for some $w\in (X\cup X^{-1})^*$. For details we refer to [2]. We illustrate the situation by the following example.

Example 2.1

Let $X = \{x,y\}$ and let $G = \{1_G,g,h,gh\}$ be the X-generated Klein four-group with ${\overline{x}}:= g$ and ${\overline{y}}:= h$. Then $\Gamma (G) = \{(1_G,x), (1_G,y), (g,x),$ $(g,y), (h,x), (h,y), (gh,x), (gh,y)\}$. Now, let e.g. $w = xy^{-1}x^{-1}\in (X\cup X^{-1})^*$. We get $w^{\prime } = x{\overline{y}}^{\,-1}y{{\overline{y}}}^{\,-1}{{\overline{x}}}^{\,-1}x{{\overline{x}}}^{\,-1} = xhyghxg$, whence $\left\langle w \right\rangle$ corresponds to

$$\begin{aligned} (\{(1_G,x),(gh,y),(h,x)\},h)\in M(G). \end{aligned}$$

On the other hand e.g. $(\{(1_G,x),(g,x),(h,y)\},g)\in M(G)$ corresponds to $\left\langle x^2 y^{-1}yx^{-1} \right\rangle$, being equal to e.g. $\left\langle x^2 y^{-1}yx \right\rangle$.

3 Canonical dual prehomomorphisms into M(G)

In M(G) the natural partial order is given by $\langle v\rangle \le \langle w\rangle$ if and only if ${\overline{v}}={\overline{w}}$ and $\Gamma (\langle w\rangle )\subseteq \Gamma (\langle v\rangle )$. The following order theoretic statements are straightforward.

Proposition 3.1

The least upper bound $\vee _{i\in I}\langle w_i\rangle$ with respect to $\le$ exists in M(G) if and only if all ${\overline{w_i}}$ are equal to a given ${\overline{w}}$, say, and

• (1) ${\overline{w}}\ne 1_G$ and ${\overline{w}}$ is a vertex of the $1_G$ containing connected part of $\cap _{i\in I}\Gamma (\langle w_i\rangle )$, denoted by $\mathrm {cp}(\cap _{i\in I}\Gamma (\langle w_i\rangle ))$, in which case $\vee _{i\in I}\langle w_i\rangle \;\widehat{=}\;(\mathrm {cp}(\cap _{i\in I}\Gamma (\langle w_i\rangle )) ,{\overline{w}})$ or

• (2) ${\overline{w}}= 1_G$ in which case $\vee _{i\in I}\langle w_i\rangle \;\widehat{=}\;(\mathrm {cp}(\cap _{i\in I}\Gamma (\langle w_i\rangle )),1_G)$, if the latter exists, and $\vee _{i\in I}\langle w_i\rangle \;\widehat{=}\;(\emptyset ,1_G)=1_{M(G)}$ otherwise.

Note that the greatest lower bound $\wedge _{i\in I}\langle w_i\rangle$ exists in M(G) for each finite set I if and only if all ${\overline{w_i}}$ are equal to a given ${\overline{w}}$, say, in which case $\wedge _{i\in I}\langle w_i\rangle \;\widehat{=}\;(\cup _{i\in I}\Gamma (\langle w_i\rangle ), {\overline{w}})$. Note further that $\vee _{i\in I}\langle w_i\rangle$ exists if and only if the set $\{\langle w_i\rangle , i\in I\}$ has an upper bound in M(G).

Let H be an X- generated group via an injection $\varepsilon _H : X\rightarrow H\setminus \{1_H\}$. Like with M(G) we may represent the elements of H by their corresponding images [w] in $(X\cup X^{-1})^*/\ker \varphi _H$, where $\varphi _H$ denotes the unique extension of $\varepsilon _H$ to a homomorphism from $(X\cup X^{-1})^*$ onto H. A mapping $\psi : H \rightarrow M(G)$ is called a dual prehomomorphism if $([v][w])\psi \ge ([v])\psi ([w])\psi$ and $([v]^{-1})\psi =([v]\psi )^{-1}$ for all $[v],[w]\in H$, see [5]. According to [1], we call $\psi$ canonical if $([x])\psi =\langle x\rangle$ for all $x\in X$. Note that a canonical dual prehomomorphism $\psi : H \rightarrow M(G)$ always induces a generator respecting homomorphism from H onto G, given by $[w]\mapsto {\overline{w}}$, which follows from the fact that in M(G) we have that $(\Gamma (\langle v\rangle ), {\overline{v}})\le (\Gamma (\langle w\rangle ), {\overline{w}})$ implies ${\overline{v}}={\overline{w}}$ and $\psi$ respects generators. Thus H necessarily must be an extension by G. Further $(1_H)\psi = 1_{M(G)}$ since

$$\begin{aligned} (1_H)\psi = ([xx^{-1}])\psi \ge ([x])\psi ([x^{-1}])\psi =([x])\psi (([x])\psi )^{-1} =\langle x \rangle \langle x^{-1} \rangle \end{aligned}$$

which corresponds to $(\Gamma (\langle x\rangle ), 1_G)=(\{(1_G,x)\},1_G)$ and on the other hand $(1_H)\psi = ([x^{-1}x])\psi \ge \langle x^{-1}\rangle \langle x\rangle$ is corresponding to $(\{({\overline{x}}^{\,-1},x)\},1_G)$. Consequently $\Gamma ((1_H)\psi )\subseteq \{(1_G,x)\}\cap \{({\overline{x}}^{\,-1},x)\}=\emptyset$ implying $(1_H)\psi = 1_{M(G)}$.

In what follows we give a necessary and sufficient condition for M(G) to admit a canonical dual prehomomorphism $\psi : H\rightarrow M(G)$. Our condition is of an order theoretic form.

Theorem 3.2

Let G and H be groups as defined above. Then H admits a canonical dual prehomomorphism $\psi : H\rightarrow M(G)$ if and only if the following sequence of least upper bounds exists for each $[w]\in H:$

$$\begin{aligned} P_0([w])&:=\vee _{[v]=[w]}\langle v\rangle \\ P_n([w])&:=\vee _{[w_1][w_2]=[w]} P_{n-1}([w_1])P_{n-1}([w_2]), \quad n\in {\mathbb {N}}. \end{aligned}$$

Proof

Necessity: Let $\psi : H\rightarrow M(G)$ be a canonical dual prehomomorphism. Let $[w]\in H$, for some $w\in (X\cup X^{-1})^*$. Since $\psi$ is canonical we obtain $([w])\psi \ge \langle v\rangle$ for all $v\in (X\cup X^{-1})^*$ with $[v]=[w]$. Consequently $P_0([w])=\vee _{[v]=[w]}\langle v\rangle$ exists and $([w])\psi \ge P_0([w])$. Let now $[u],[v]\in H$ with $[u][v]=[w]$. Then $([w])\psi = ([u][v])\psi \ge ([u])\psi ([v])\psi \ge P_0([u])P_0([v])$. Consequently $P_1([w])=\vee _{[u][v]=[w]}(P_0([u])P_0([v]))$ exists and $([w])\psi \ge P_1([w])$. Continuing this process we see that all $P_n([w]), n\in {\mathbb {N}}_0$ exist.

Sufficiency: Let the condition in the assumption of Theorem 3.2 be satisfied. Note that $\{P_n([w])\}_{n\in {\mathbb {N}}_0}$ is increasing and will be constant after a finite number of steps, for each $[w]\in H$, since all occurring graphs are finite. Let $P([w]):=\lim \limits _{n \rightarrow \infty } P_n([w]), [w]\in H$. We show that the mapping $\psi : [w]\mapsto P([w])$ defines a canonical dual prehomomorphism. Let $[u],[v]\in H$. It follows $P_1([uv])\ge P_0([u])P_0([v])$, $P_2([uv])\ge P_1([u])P_1([v])$, $\ldots$, $P_n([uv])\ge P_{n-1}([u])P_{n-1}([v])$, $\ldots$ which after a finite number of steps gives $P([uv])\ge P([u])P([v])$. Further $P([w]^{-1})=(P([w]))^{-1}$, since $\langle u\rangle \vee \langle v\rangle$ exists if and only if $\langle u\rangle ^{-1}\vee \langle v\rangle ^{-1}$ exists in which case $\langle u\rangle ^{-1}\vee \langle v\rangle ^{-1} = (\langle u\rangle \vee \langle v\rangle )^{-1}$. This fact holds in any inverse semigroup S and easily follows from $s\le t \Leftrightarrow s^{-1}\le t^{-1}$, $s,t\in S$. Finally $\psi$ is canonical since from $\Gamma (P([x]))\subseteq \Gamma (\langle x\rangle )$ we infer $\Gamma (P([x]))=\Gamma (\langle x\rangle )$, whence $P([x])= \langle x\rangle$. $\square$

Note that the above defined mapping P is the least possible canonical dual prehomomorphism with respect to the pointwise order of mappings, since in the necessity proof of Theorem 3.2 we have $([w])\psi \ge P([w]), [w]\in H$.

Corollary 3.3

In case $P_0([w])\;\widehat{=}\;(\cap _{[u]=[w]}\Gamma \langle u\rangle ,{\overline{w}})\in M(G)$, for all $[w]\in H$, it follows $P_0([w])=P_n([w])$, for all $n\in {\mathbb {N}}$, whence $([w])\psi =P_0([w])$ defines a canonical dual prehomomorphism $\psi : H \rightarrow M(G)$.

Proof

Under the assumptions we obtain for arbitrary $[w_1], [w_2]\in H$ with $[w_1][w_2]=[w]$

$$\begin{aligned} P_0([w_1])P_0([w_2]) &\,\,\widehat{=} \,(\cap _{[u_1]=[w_1]} \Gamma (\langle u_1\rangle )\cup {\overline{w_1}}\cap _{[u_2]=[w_2]} \Gamma (\langle u_2\rangle ),{\overline{w}})\\ & \le (\cap _{[u]=[w]} \Gamma (\langle u\rangle ),{\overline{w}}) \\ &\,\,\widehat{=}\,P_0([w]), \end{aligned}$$

since $\cap _{[u_1]=[w_1]} \Gamma (\langle u_1\rangle )\cup {\overline{w_1}}\cap _{[u_2]=[w_2]} \Gamma (\langle u_2\rangle )\supseteq \cap _{[u]=[w]} \Gamma (\langle u\rangle ).$ Thus we have

$$\begin{aligned} P_1([w])=\vee _{[w_1][w_2]=[w]} (P_0([w_1])P_0([w_2]))\le P_0([w])\le P_1([w]), \end{aligned}$$

whence $P_1([w])=P_0([w])$ follows. We conclude by induction

$$\begin{aligned} P_0([w])=P_1([w])=P_2([w])=\cdots =P([w]), \end{aligned}$$

proving the assertion. $\square$

Example 3.1

Let G be any X-generated group and let H be the free group on X. Then for any $[w] \in H$ we have $P_0([w]) \,\widehat{=} \, (\Gamma (\left\langle r(w) \right\rangle ),{\overline{w}}) \in M(G)$ where r(w) is the reduced word associated to $\left[ w \right]$.

Example 3.2

Let G be the $\{x\}$-generated cyclic group of order n and let H be the $\{x\}$-generated cyclic group of order 2n. Inspecting $\Gamma (G)$ which is an n-cycle, we directly see

$$\begin{aligned} \cap _{[w]=[x^k]}\Gamma (\langle w \rangle ) = \Gamma (\langle x^k \rangle ), \quad 1 \le k \le n \end{aligned}$$

and

$$\begin{aligned} \cap _{[w]=[x^l]}\Gamma (\langle w \rangle ) = \Gamma (\langle x^{l-2n} \rangle ), \quad n \le l \le 2n. \end{aligned}$$

In particular we have $\cap _{[w]=[x^{2n}]}\Gamma (\langle w \rangle ) = \emptyset$, since $[\emptyset ] = [x^{2n}]$ corresponds to $1_{H}$. Hence $\psi :H \rightarrow M(G)$ may be defined by $([x^k])\psi = \langle x^k \rangle , 1 \le k \le n, ([x^l]) \psi = \langle x^{l-2n} \rangle , n< l <2n,$ and $([x^{2n}]) \psi = \langle \emptyset \rangle \;\widehat{=} \;(\emptyset , 1_{G})=1_{M(G)}$, cf. ( [2], Theorem 19).

To check whether a given extension H by a group G satisfies the condition of Theorem 3.2 it is crucial to determine $\cap _{[v]=[w]}(\Gamma (\langle v \rangle ))$ for any $[w] \in H$. In what follows we describe a way of doing that for finite H and G which might be implemented on a computer. We start to determine a finite subset T of $(X \cup X^{-1})^{*}$ satisfying the following property: For each $w \in (X \cup X^{-1})^{*}$ there is $v \in T$ such that $[w] = [v]$ and $\Gamma (\langle v \rangle ) \subseteq \Gamma (\langle w \rangle )$. To compute such a set T we describe a simple algorithm which directly implements the defining property of T.

• (0) Put the identity element of $(X \cup X^{-1})^{*}$ into T.

• (1) For T, constructed so far, construct a superset $T^{\prime }$ of T in the following way: Put all elements of T into $T^{\prime }$. List the elements of $T\times X \times \{-1,1\}$ and check for each $(w, x ,\varepsilon )$ in $T\times X \times \{-1,1\}$ if there is $u \in T$ such that $[u] = [wx^\varepsilon ]$ and $\Gamma (\langle u \rangle ) \subseteq \Gamma (\langle wx^\varepsilon \rangle )$. If the answer for a given $(w, x ,\varepsilon )$ is yes, go to the next triple in the list. If the answer is no, put $wx^\varepsilon$ into $T^{\prime }$ and go to the next triple in the list.

• (2) If T is a proper subset of $T^{\prime }$, as constructed in (1), take $T^{\prime }$ as new T and start (1) again. If $T = T^{\prime }$ the algorithm stops.

Note that since H and M(G) are finite, the computation stops after a finite number of steps. To see that in the end T has the required property, we note that if a word $w^{\prime }$ is dropped in (1) of the above algorithm because $[w^{\prime }] = [u]$ with $\Gamma (\langle u \rangle ) \subseteq \Gamma (\langle w^{\prime } \rangle )$ for some $u\in T$, then for each word $w^{\prime }v, v\in (X \cup X^{-1})^{*}$ we have $[w^{\prime }v] = [uv]$ with $\Gamma (\langle uv \rangle ) \subseteq \Gamma (\langle w^{\prime }v \rangle )$, where uv is in T or has been dropped earlier in (1), i.e. there is some $u^{\prime }\in T$ such that $[uv] = [u^{\prime }]$ and $\Gamma (\langle u^{\prime } \rangle ) \subseteq \Gamma (\langle uv \rangle )$, whence $[w^{\prime }v] = [u^{\prime }]$ and $\Gamma (\langle u^{\prime } \rangle ) \subseteq \Gamma (\langle w^{\prime }v \rangle )$. Consequently the final set T satisfies the property that for each word w in $(X \cup X^{-1})^{*}$ there is a word u in T such that $[w] = [u]$ and $\Gamma (\langle u \rangle ) \subseteq \Gamma (\langle w \rangle )$. Now for a given $[w] \in H$ we get

$$\begin{aligned} \cap _{[v]=[w]} \Gamma (\langle v \rangle )= \cap _{[u]=[w]} \Gamma (\langle u \rangle ), \end{aligned}$$

where $v \in (X \cup X^{-1})^{*} , u \in T$, and in case ${\overline{w}} \ne 1_G$ we have to check whether the right hand intersection contains a connected subgraph with vertices $1_G$ and ${\overline{w}}$, to see whether $P_0([w])$ exists. Note that in case ${\overline{w}} = 1_G$, $P_0([w])$ always exists. If for some $[w] \in H$, $P_0([w])$ does not exist, the algorithm stops. If all $P_0([w])$, $[w] \in H$ exist, we check whether for each $[w] \in H$ $P_1([w]) = \vee _{[w_1][w_2]=[w]}(P_0([w_1])P_0([w_2]))$ exists, by going through all |H| factorisations of [w]. If $P_1([w])$ does not exist for some $[w] \in H$, the algorithm stops. In the other case we continue, checking whether $P_2([w])$ exists, and so on. After a finite number of computations we end up with $n_0 \in {\mathbb {N}}$ such that either $P_{n_0}([w])$ does not exist for some $[w] \in H$, in which case H does not satisfy the conditions of Theorem 3.2 , or $P_{n_0}([w]) = P_{n_0+1}([w])$ for all $[w] \in H$. The latter must be the case since for each $[w] \in H$ the sequence $\{P_n([w])\}_{n \in {\mathbb {N}}_0}$ is decreasing whence eventually constant, since all occurring graphs are finite. Further H is finite. We then have $P_{n_0}([w]) = P_k([w])$ for all $k \ge n_0$, $[w] \in H$. Thus H satisfies the conditions of Theorem 3.2.

Even for a small finite noncyclic X-generated group G, an X-generated group H admitting a canonical dual prehomomorphism $\psi :H \rightarrow M(G)$ might be large. The following theorem points into this direction.

Theorem 3.4

Let $G=\{1_G, g, h, gh\}$ be the $\{x, y\}$-generated Klein four-group with respect to ${\overline{x}}=g, {\overline{y}}=h$. Then any X-generated group H which admits a canonical dual prehomomorphism $\psi :H \rightarrow M(G)$ must be of exponent 6 at least.

Proof

We show that the $\{x, y\}$-generated Burnside group of exponent 4, B(2; 4), does not admit a suitable $\psi :B(2;4) \rightarrow M(G)$. Assume that $\psi$ exists. Note first that in B(2; 4) we have $[xyx^2yx^{-1}]=[x^{-1}y^{-1}x^2y^{-1}x]$, since

$$\begin{aligned}{}[xyx^2yx^{-1}]&=[xyx^2yx^3] \\&=[(xyx^2yx^2)x] \\&=[x^{-1}y^{-1}yx^2yx^2yx^2x]\\&=[x^{-1}y^{-1}(yx^2)^{-1}x]\\&=[x^{-1}y^{-1}x^{-2}y^{-1}x]\\&=[x^{-1}y^{-1}x^2y^{-1}x] =:[u]. \end{aligned}$$

We get $\langle xyx^2yx^{-1} \rangle \le ([xyx^2yx^{-1}])\psi =([x^{-1}y^{-1}x^2y^{-1}x])\psi \ge \langle x^{-1}y^{-1}x^2y^{-1}x \rangle$, whence $\Gamma ([u]\psi ) \subseteq \Gamma (\langle xyx^2yx^{-1} \rangle ) \cap \Gamma (\langle x^{-1}y^{-1}x^2y^{-1}x \rangle )$.

Since the intersection of both graphs does not contain a connected subgraph having at least one edge and vertex $1_G$, we conclude that $\Gamma ([u]\psi ) = \emptyset$, whence $([u])\psi =1_{M(G)}$. We infer

$$\begin{aligned} ([x^2y^{-1}x])\psi&= ( [ yxx^{-1}y^{-1}x^2y^{-1}x ] ) \psi \\&\ge ([yx]) \psi \underbrace{( [ x^{-1}y^{-1}x^2y^{-1}x ] ) \psi }_{1_{M(G)}} =( [yx] ) \psi \ge \langle yx \rangle , \end{aligned}$$

and on the other hand $([x^2y^{-1}x]) \psi \ge \langle x^2y^{-1}x \rangle$ which means

$$\begin{aligned} \Gamma ( ([x^2y^{-1}x])\psi ) \subseteq \Gamma (\langle yx \rangle ) \cap \Gamma (\langle x^2y^{-1}x \rangle ) \end{aligned}$$

with contradiction, since the intersection on the right hand side does not contain a connected subgraph with vertices $1_G$ and ${\overline{x^2y^{-1}x}} = gh$.

$\square$

It is an open question whether the finite group B(2; 6) admits a canonical dual prehomomorphism into M(G) with G the Klein four-group, or a contradiction can be achieved following the pattern in the proof of Theorem 3.4. It is also an open question whether the group $G^{U}$, as defined in [1], with U the variety of all groups of exponent $n=3$, respectively $n=4$, admits a suitable mapping $\psi :G^{U} \rightarrow M(G)$ in this case. In our setting $G^{U}$ may be represented by $FG(\{x,y\})_{/\equiv }$, where $\equiv$ is the congruence on the free group $FG(\{x,y\})$ generated by the relators $w^{3}=1$, respectively $w^{4}=1$, where ${\overline{w}}=1_{G}$, $w \in FG(\{x,y\})$. Since, by construction in [1], $G^{U}$ is a subgroup of a semidirect product of the finite groups B(8; 3), respectively B(8; 4) by G, it is finite. Obviously B(2; 4) is a homomorphic image of $G^{U}$ in case $n=4$. However B(2; 4) itself is not of the form $G^V$ for some group variety V, since the only possible choice of such V would be the variety of elementary Abelian 2-groups. Only if V has exponent 2, the group $G^V$ has exponent $2\cdot 2 = 4$. But in this case $G^V$ is a subgroup of a semidirect product of the free elementary Abelian 2-group of rank 8 by G whence $|G^V|< 2^8\cdot 2^2 = 2^{10} < 2^{12} = |B(2;4)|$. Note in particular that $G^{U}$ has exponent 6 in case $n=3$, and exponent 8 in case $n=4$. Anyway it follows from [1], Proposition 4.4., referring to a remark of V. Guba, that $\psi :G^{U} \rightarrow M(G)$ exists if U is the variety of all groups of sufficiently large odd exponent n.

We continue our considerations with a theorem which also follows from a result of Szakács [6]. For sake of completeness we give an elementary direct proof.

Theorem 3.5

Let G be an X-generated noncyclic group, and let H be a generator respecting X-generated extension by G such that the homomorphism $H \rightarrow G$, defined by $[w] \mapsto {\overline{w}}$ has a nontrivial Abelian kernel K. Then there is no canonical dual prehomomorphism $\psi :H \rightarrow M(G)$.

Proof

We show first that under the assumptions $\Gamma (G)$ contains a subgraph consisting of two disjoint cycles connected by a path, of the form

where $u_1, u_2, v, z_1, z_2$ are nonempty words in $(X \cup X^{-1})^{*}$, labeling the respective paths.

Assume first that there is $y \in X$ such that ${\overline{y}}$ has finite order $m \ge 2$. Since G is noncyclic there is $x \in X$ such that ${\overline{x}} \ne {\overline{y}}^n$, for all $n \in {\mathbb {N}}$. Consequently, by use of the words $u_1=z_1=y, u_2=y^{1-m}, v=x, z_2=y^{m-1}$, we may define a graph which consists of two cycles with vertex sets $A= \{ 1_G, {\overline{y}}, \ldots , {\overline{y}}^{m-1} \}$ and $B= \{ {\overline{yx}}, {\overline{yxy}}, \ldots , {\overline{yx}}\, {\overline{y}}^{m-1} \}$ connected by the edge $({\overline{y}},x)$. Since A is a subgroup of G and $B={\overline{yx}}A$, with ${\overline{yx}} \notin A$ by assumption, we obtain $A \cap B = \emptyset$.

Assume now that there is $x\in X$, such that ${\overline{x}}$ has infinite order. Since K is nontrivial there is a nonempty reduced word $w=y_1\ldots y_m$, $m\ge 2$, with $y_i\in X \cup X^{-1}$, $1\le i\le m$, such that ${\overline{w}}=1_G$, and $\Gamma (\langle w \rangle )$ forms a cycle. Let $u_1=y_1=z_1, u_2=(y_2 \cdots y_m)^{-1}$, $z_2=y_2 \cdots y_m$, and $v^{\prime }=x^n$, where n is such that ${\overline{y_1}}\,{\overline{x}}^na\ne b$ for all a, b in the set $A=\{1_G,{\overline{y_1}},\ldots ,{\overline{y_1 \cdots y_{m-1}}}\}$. Such n exists, since the equality ${\overline{y_1}}\,{\overline{x}}^ka= b$ can only hold for at most one $k\in {\mathbb {N}}$ by the assumption that ${\overline{x}}$ has infinite order, and the set A, whence $A\times A$, is finite. We may use $u_1,u_2,v^{\prime },z_1,z_2$ to define a graph which consists of the two disjoint cycles with vertex sets $A=\{1_G,{\overline{y_1}},\ldots ,{\overline{y_1 \cdots y_{m-1}}}\}$ and $B=\{{\overline{y_1}}\,{\overline{x}}^n,{\overline{y_1}}\,{\overline{x}}^n{\overline{y_1}},\ldots ,{\overline{y_1}}\,{\overline{x}}^n{\overline{y_1 \cdots y_{m-1}}}\}$, connected by the path with initial vertex ${\overline{y_1}}$ labeled by $v^{\prime }=x^n$. Let $p,q\in \{1,\ldots ,n\}$ such that p is the least element with ${\overline{y_1}}\,{\overline{x}}^p\notin A$ for all k, $p\le k\le n$, and such that q is the least element with ${\overline{y_1}}\,{\overline{x}}^q\in B$. Then the path with initial vertex ${\overline{y_1}}\,{\overline{x}}^{p-1}$ and label $v=x^{q-p+1}$ connects the cycles with vertex sets A and B precisely as shown in the graph above. We conclude

$$\begin{aligned} \langle u_1vz_1z_2v^{-1}u_1^{-1} \rangle \vee \langle u_2vz_1z_2v^{-1}u_2^{-1} \rangle = 1_{M(G)}. \end{aligned}$$

On the other hand we obtain

$$\begin{aligned}{}[u_1vz_1z_2v^{-1}u^{-1}_1]&= [u_1vz_1z_2v^{-1}u_1^{-1}u_2u_2^{-1}] \\&= [u_1u_1^{-1}u_2vz_1z_2v^{-1}u_2^{-1}], \text { since } [u_1^{-1}u_2], [vz_1z_2v^{-1}] \in K \\&= [u_2vz_1z_2v^{-1}u_2^{-1}]. \end{aligned}$$

Hence for any canonical dual prehomomorphism $\psi : H \rightarrow M(G)$ we get

$$\begin{aligned} \left( [u_1vz_1z_2v^{-1}u_1^{-1}] \right) \psi&= \left( [u_2vz_1z_2v^{-1}u_2^{-1}] \right) \psi \\&\ge \langle u_1vz_1z_2v^{-1}u_1^{-1} \rangle \vee \langle u_2vz_1z_2v^{-1}u_2^{-1} \rangle = 1_{M(G)}, \end{aligned}$$

whence $\left( [u_1vz_1z_2v^{-1}u_1^{-1}] \right) \psi = 1_{M(G)}$.

By the rule $\left( [w_1w_2] \right) \psi = 1_{M(G)} \Rightarrow ([w_2])\psi \ge ([w_1^{-1}])\psi \Rightarrow ([w_2^{-1}])\psi \ge ([w_1])\psi$, since $\psi$ respects inverses, we obtain with $[w_1] = [u_1vz_1]$ and $[w_2] = [z_2v^{-1}u_1^{-1}]$ that $([u_1vz_2^{-1}])\psi \ge ([u_1vz_1])\psi \ge \langle u_1vz_1 \rangle$, which together with $([u_1vz_2^{-1}])\psi \ge \langle u_1vz_2^{-1} \rangle$ leads to a contradiction. Note in particular that $\langle u_1vz_1 \rangle \vee \langle u_1vz_2^{-1} \rangle$ does not exist. $\square$

Note that in case K is trivial in Theorem 3.5, i.e. $K= \left\{ 1_H\right\}$, we have that H is isomorphic to G via the homomorphism induced by the mapping $[x] \mapsto {\overline{x}}$.

It is shown in [2], see also Example 3.2, that for any $\{x\}$-generated cyclic group G of order n there is a canonical dual prehomomorphism $\psi$ from the $\{x\}$-generated cyclic group H of order 2n into M(G). Clearly the homomorphism $[w] \mapsto {\overline{w}}$ has Abelian kernel. If we regard, however, G as an e.g. $\{x,y\}$-generated group where ${\overline{y}}={\overline{x}}^2$, say, $n\ge 3$, then the assertion of Theorem 3.5 remains true, although G is cyclic, as the following example shows for $n=3$.

Example 3.3

Let $G = \{1_G,g,g^2\}$ be the three element cyclic group generated by $X = \{x,y\}$ with respect to ${\overline{x}}=g,{\overline{y}}=g^2$. The Cayley graph $\Gamma (G)$ looks as follows:

Assume $\psi$ exists for some $\{x,y\}$-generated generator preserving group extension H by G , where $[w] \rightarrow {\overline{w}}$ has Abelian kernel. It follows

$$\begin{aligned}{}[y^{-1}xy^2]&= [x(x^{-1}y^{-1})(xy)y] \\&= [x(xy)(x^{-1}y^{-1})y], \text { since } {\overline{xy}} = {\overline{x^{-1}y^{-1}}} = 1_{G} \\&= [x^2yx^{-1}]. \end{aligned}$$

We obtain $\left( \left[ y^{-1}xy^2 \right] \right) \psi \ge \langle y^{-1}xy^2 \rangle ,\langle x^2yx^{-1} \rangle$

$\Rightarrow \Gamma \left( \left( \left[ y^{-1}xy^2 \right] \right) \psi \right) \subseteq \Gamma \left( \langle y^{-1}xy^2 \rangle \right) \cap \Gamma \left( \langle x^2yx^{-1} \rangle \right)$

$\Rightarrow \Gamma \left( \left( \left[ y^{-1}xy^2 \right] \right) \psi \right) = \emptyset \Rightarrow \left( \left[ y^{-1}xy^2 \right] \right) \psi = 1_{M(G)}$.

We infer

$$\begin{aligned} \left( [x]\right) \psi&= \left( [y(y^{-1}xy^2)y^{-2}] \right) \psi \\&\ge \left( [y]\right) \psi \underbrace{\left( [y^{-1}xy^2]\right) \psi }_{1_{M(G)}} \left( [y^{-2}]\right) \psi \\&= \left( [y]\right) \psi \left( [y^{-2}]\right) \psi \ge \langle yy^{-1}y^{-1} \rangle , \end{aligned}$$

which means together with $([x])\psi = \langle x \rangle$ a contradiction.

Note that $\Gamma (G)$ contains a forbidden minor in the sense of Szakács, namely

In particular $\Gamma (\langle x\rangle )$ is a breaking path in her terminology.