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Noncyclic graph of a group. (English) Zbl 1131.20016

In two papers published as Technical Reports at the Rose-Hulman Institute of Technology the idea of a cyclicizer of an element was introduced. For an element $x\in G$, ${\text{Cyc}}_{G}\left(x\right)=\left\{y\mid 〈x,y〉$ is cyclic $\right\}$. For a subset $X$, ${\text{Cyc}}_{G}\left(X\right)$ is defined to be the intersection of the cyclicizers of the elements of $X$. Note that ${\text{Cyc}}_{G}\left(G\right)=\text{Cyc}\left(G\right)$, the cyclicizer of $G$. Trivially $\text{Cyc}\left(G\right)\le Z\left(G\right)$.

The paper is organised in eight sections. The second contains some basic facts about cyclicizers and discusses some results about non-locally-cyclic groups with $\text{Cyc}\left(G\right)\ne 1$. The authors introduce the non-cyclic graph of a group $G$, ${\Gamma }\left(G\right)$, this is defined to have vertices $G\setminus \text{Cyc}\left(G\right)$ with two vertices $x$ and $y$ joined if $〈x,y〉$ is not cyclic. In Section 3 they show that ${\Gamma }\left(G\right)$ is connected and that the diameter is at most 3, and the example they give of a group with diameter 3 is a direct product of the symmetric group of degree 6 and the cyclic group of order 6. It is curious that such a restriction on diameters is not uncommon in graphs defined by groups.

There is an alternative graph, the non-commuting graph defined to have vertices $G\setminus Z\left(G\right)$ with two vertices $x$ and $y$ joined if $\left[x,y\right]\ne 1$, this is a subgraph of the non-cyclic graph. The non-commuting graph was studied by B. H. Neumann [J. Aust. Math. Soc., Ser. A 21, 467-472 (1976; Zbl 0333.05110)]. In this paper he showed that the graph contains no infinite cliques if and only if $G/Z\left(G\right)$ is cyclic. They show that the equivalent result holds for the non-cyclic graph, that is ${\Gamma }\left(G\right)$ has no cliques if and only if $G/\text{Cyc}\left(G\right)$ is cyclic.

The last two sections consider the interesting problem of when ${\Gamma }\left(G\right)$ uniquely determines the group, though even the general question of the order for finite groups is not answered. It should be noted that this only makes sense for groups which are not locally cyclic. They give a number of results which discuss special cases. One of which leads to a conjecture of Goormaghtigh on numbers of the form $\left({x}^{n}-1\right)/\left({x}^{m}-1\right)$.

##### MSC:
 20D60 Arithmetic and combinatorial problems on finite groups 05C25 Graphs and abstract algebra