Suppose $G$ is a finite group. Let’s call an element $g \in G$ primitive, iff $\forall h \in G$, such that $\exists n \in \mathbb{N}, h^n = g$ it is true, that $\langle h \rangle = \langle g \rangle$. Suppose $P(G)$ is the set of all primitive elements of $G$. It is not hard to see, that $\langle P(G) \rangle = G$ and moreover $\forall g \in G \exists h \in P(G) g \in \langle h \rangle$.

Now, suppose the cycle graph of $G$ is an undirected graph $Cyc(G) = (V, E)$, such that the set of all vertices $V = G$ and the set of all edges is $E = \{(g^n, g^{n+1})|g \in P(G) n \in \mathbb{N}\}$.

Now, let’s consider automorphism groups of cycle graphs. I have managed to find them for the following groups: $$Aut(Cyc(E)) \cong E$$ $$Aut(Cyc(C_2)) \cong C_2$$ $$Aut(Cyc(C_2^n)) \cong S_{2^n - 1}, \forall n > 1$$ $$Aut(Cyc(C_m)) \cong D_m, \forall m > 2$$ $$Aut(Cyc(C_m^n)) \cong C_2^{\frac{m^n - 1}{m - 1}} \times S_{\frac{m^n - 1}{m - 1}}, \forall m > 2, n > 1$$ $$Aut(Cyc(D_m)) \cong C_2 \times S_m, \forall m > 2$$ $$Aut(Cyc(Q_8)) \cong C_2 \times S_6$$

The other finite groups seem to have too complicated cycle graphs to find all their automorphisms manually.

As you may have noticed, among the groups above, only $E$ and $C_2$ satisfy the condition $G \cong Aut(Cyc(G))$. The question is - are there any other such finite groups?

This question can also be reworded as: «Does $G \cong Aut(Cyc(G))$ imply $|G| \leq 2$

If a counterexample exist, it is not a direct product of isomorphic cyclic groups, not a dihedral group and not the quaternion group. However, I do not know, how to deal here with groups, that do not fall into any of those classes.


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