Errata to: A New Graph over Semi-Direct Products of Groups

TL;DR

This paper investigates a proposed graph structure on groups with semi-direct product decompositions, revealing it to be ill-defined and exploring reasons for this inconsistency.

Contribution

It critically analyzes a proposed graph on semi-direct product groups, identifying issues and providing insights into its ill-defined nature.

Findings

The graph mma(G) is ill-defined for groups with semi-direct product structure.

The paper identifies specific causes for the discrepancies in the graph's definition.

Abstract

The goal of the paper "A new graph over semi-direct products of groups" is to define a graph \Gamma(G) on a group G when G splits over a normal subgroup. We demonstrate herein that the graph is ill-defined. We also attempt to ascertain causes for the discrepancies.

Equations

$$\mathrm{DS}(\Gamma (D_8))=\{1,1,1,4,4,4,4,7\}\text{and}\mathrm{DS}(\Gamma (G))=\{1,2,2,2,4,4,4,7\}.$$

Full text

**MSC: **Primary 20A99, 05C12

1 Background

In group theory, a familiar theme is the application of group invariants that can be used to demonstrate two groups are not isomorphic. In this vein, placing a graph structure on groups and showing corresponding graph invariants differ shows the graphs are distinct, hence the underlying groups are nonisomorphic. The goal of the paper [3] is such a graph. There however is a huge problem with the paper; the graph it presents is ill-defined. In fact, the paper explicitly presents two isomorphic groups then computes distinct graph invariants thereof. Hence, the purpose of this paper is to demonstrate the graph is in fact ill-defined.

The graph in question is described as being on groups $G$ which are split extensions $G=K\rtimes A$. Letting $[X;\mathbf{r}]$ and $[Y;\mathbf{s}]$ be respectively presentations for $K$ and $A$ (with $X$ the generators and r the relations), the graph $\Gamma(G)$ has vertex set $G$ and (quoting the paper directly) “…the edge set $E$ is obtained by the following steps:

(I) Each of the vertices in this graph must be adjoined to the vertex $1_{G}$ (except $1_{G}$ itself since the graph is assumed to be simple).

(II)(i) For any two vertices $w_{1}=x_{1}^{\varepsilon_{1}}x_{2}^{\varepsilon_{2}}\dots x_{m}^{\varepsilon_{m}}$ and $w_{2}=y_{1}^{\delta_{1}}y_{2}^{\delta_{2}}\dots y_{n}^{\delta_{n}}$ (where $n\geq 2$, $\varepsilon_{i}$ and $\delta_{i}$ are integers) and for all $x_{i},y_{j}\in X\cup Y~{}(1\leq i\leq m,1\leq j\leq n)$, if $x_{i}\not=y_{j}$, then $w_{1}$ is adjoined to the $w_{2}$ (shortly, $w_{1}\sim w_{2}$).

(ii) As a consequence of (i), for any two vertices $w_{1}=x_{i}{}^{k}$ and $w_{2}=x_{j}{}^{t}~{}(1\leq i,j\leq n,~{}i\not=j$, and $k,t$ are integers), we can directly take $w_{1}\sim w_{2}$. However, to adjoin $w_{1}$ and $w_{2}$ while $i=j$, it must be $k\not=t$.”

Clearly, whether or not an edge connects two vertices is strongly tied to the presentation. To mitigate this dependence, the article states “…all elements $z_{i}~{}(i=1,2,\dots,k)$ in the generating set $X\cup Y$ of $G$ will be formed as $z_{i}\not=z_{1}^{\varepsilon_{1}}z_{2}^{\varepsilon_{2}}\dots z_{k}^{\varepsilon_{k}}$, where $k\geq 2$ according to the Normal Form Theorem (NFT) (see [11]).” (The article’s reference [11] is our reference [2].)

2 Counterexample

In trying to digest the above definition (wherein [math] exponents may or may not be allowed, wherein order of multiplicands is not specified, wherein (II)(ii) claims the adjacency of $x_{i}{}^{k}$ and $x_{j}{}^{t}$ when $i\not=j$ follows from (II)(i) while (II)(i) has a hypothesis that there be at least $n\geq 2$ generators in at least one of the vertices, etc.), this article’s author turned to the examples presented in [3] to see what interpretations were used there; specifically, Examples 2.9 and 2.10. Example 2.9 is given as the dihedral group $D_{8}$ viewed as the split extension of the cyclic $C_{4}$ being acted upon by $C_{2}$ (the action, of course, being inversion). Meanwhile, Example 2.10 is the group $G$ defined as the split extension of the Klein $4$-group $\mathcal{V}_{4}$ being acted upon by $C_{2}$. Since “…the homomorphism $\varphi$ will always be not identity $id_{G}$ unless stated otherwise,” we conclude this is a nontrivial action, in which case $C_{2}$ exchanges two of the nontrivial members, while fixing the third. The article [3] calculates the respective degree sequences as

[TABLE]

In point of fact, $D_{8}$ and $G=\mathcal{V}_{4}\rtimes C_{2}$ are isomorphic. To see this, we invoke (23.4) on page 107 of [1], which states:

Let $G$ be a nonabelian group of order $p^{n}$ with a cyclic subgroup of index $p$. Then $G\cong\text{Mod}_{p^{n}}$, $D_{2^{n}}$, $SD_{2^{n}}$, or $Q_{2^{n}}$.

The paragraph immediately preceding this result states the modular group $\text{Mod}_{8}=D_{8}$, while two paragraphs further back states the semidihedral group $SD_{2^{n}}$ is only defined for $n\geq 4$. Now, consider an arbitrary nonabelian group $H$ of order $8$. Since every group of exponent $2$ is necessarily abelian222Proof: $xxyy=x^{2}y^{2}=1=(xy)^{2}=xyxy$ implies $xy=yx$ over all $x,y$ in the group., $H$ must have an element of order $4$. It therefore has a cyclic subgroup of index $2$ and thus satisfies the theorem. Consequently, there are exactly $2$ nonabelian groups of order $8$: the dihedral group $D_{8}$ and the quaternion group $Q_{8}=\{\pm 1,\pm i,\pm j,\pm k\}$. The only involution in $Q_{8}$ is $-1$. Therefore, whenever a nonabelian group $H$ of order $8$ has more than one element of order $2$, then necessarily $H$ is dihedral. The subgroup $\mathcal{V}_{4}$ of $G=\mathcal{V}_{4}\rtimes C_{2}$ already contains 3 involutions; $G$ is therefore not quaternion, and is then necessarily dihedral.

3 The Errors

Given that the same group yielded distinct graphs, where are the logical flaws in [3]? One place where fault can be found is in the quoting of the Normal Form Theorem (NFT). Although not mentioned in the article [3], the open letter makes explicit that the NFT being referenced is found on page 31 of [2]333A look at [2]’s index reveals four NFTs; there was thus need for external verification as to which was being used.. This NFT is in regards to an amalgamated free product, not to a semidirect product. Another issue at point, which comes up in the given examples above, is that a semidirect factorization of groups is not unique; a group $G$ can split over nonisomorphic, normal subgroups $N_{1},N_{2}$ even with $|N_{1}|=|N_{2}|$. In short, there is no clear way to adjust this graph’s definition to allow the resulting object to reflect the underlying group’s structure.