I was perusing the cycle graphs for small groups on Wikipedia and something bothers me: is the cycle graph of a finite group actually unique (up to isomorphism)?
For example, if there are any cyclic subgroups of order $5$, the cycle graph is drawn by picking one primitive generating element $a$, and drawing a $5$-cycle in the graph between $e, a, a^2, a^3, a^4$. But a priori, this means the graph will depend on the choice of $a$. Can this result in multiple cycle graphs for the same group? It doesn't seem obvious that these different graphs would be isomorphic in general.
Definition: A cycle graph of a finite group $G$ is a simple undirected graph defined as follows: first, the vertex set of the graph is taken to be the set of elements $g \in G$. Then, for each maximal cyclic subgroup of $G$ (cyclic subgroup not fully contained in a larger cyclic subgroup), pick a generator $a$ of the subgroup, and draw undirected edges $e \to a \to a^2 \to a^3 \to \cdots \to a^{k-1} \to a^k = e$ (ignoring any duplicate edges), where $k$ is the order of the subgroup.
My question is whether the cycle graph of $G$ is unique up to isomorphism, regardless of the choices of generator for each maximal cyclic subgroup. Notice that for the purposes of this question, the graph is completely unlabeled — the original vertex labels (elements of the group) are ignored, and edges are not labeled with the cyclic subgroup they correspond to.
Strangely, I can't find a previous thread on this:
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Do cycle graphs determine groups up to isomorphism? asks the converse question of whether the cycle graph uniquely determines the group; Chris Culter asks my question in the comments but is unanswered.
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How in general does one construct a cycle graph for a group? asks for how to construct the cycle graph, but the top answer suffers from the same problem that the choice of primitive element for a cycle is not unique.
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I also searched on Google Scholar. I found an interesting paper, The Cyclic Graph of a Finite Group (Ma, Wei, Zhong), but it defines the cycle graph differently, where $x, y$ share an edge if $\langle x, y \rangle$ is cyclic. In this definition the graph is clearly unique. This also seems to me a much more sensible definition, but I don't have an example where Wikipedia's definition actually leads to ambiguity in the resulting graph, up to isomorphism.
Best Answer
In general, the isomorphism type of the cycle graph does depend on the choices of generators. Here is an example (inspired by Hagen von Eitzen's comment). Let $G=C_{10}\times C_2$, with the first factor generated by $a$ and the second generated by $b$. This group has three maximal subgroups, all of order $10$. So the edges of the cycle graph can be decomposed into three $10$-cycles. The five elements in the characteristic subgroup $H$ of order $5$ are contained in every subgroup of order $10$, so they have valency $6$, while the elements of order $2$ and $10$ are contained in a unique subgroup of order $10$, so they have valency $2$.
Now, for some choices of generators, for example $\{a,ab,a^6b\}$, elements of $H$ have exactly two elements at distance $2$, which happen to be in $H$, and there are three $2$-paths between them, as in https://en.wikipedia.org/wiki/File:GroupDiagramMiniC2C10.png
For some other choices of generators, for example $\{a,ab,a^2b\}$, this is not the case. (In this case, there are four vertices at distance $2$, some with a unique path, some with two paths.)