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Recently, when I was working with Cayley graphs, I faced up with a special group. The original group is as follows: $$G:=<a,b,c|ab=ba,a^{10}=cbc^{-1}>.$$ We can show that this group can be rewrite as …

There is a related paper for the general case, where all elements (nontrivial) has prime power order. Although, this paper does not answer your question in generally, but it has some good techniques f …

As a first-round answer and some detailed about this problem: Let $G$ be a $p-$group of odd order such that every abelian normal subgroup has at most $k$ generators, then every subgroup of $G$ has at …

As dear Derek Holt said, the answer is yes. These two references are related to this problem: $1)$ "The Faithful Linear Representation of Least Degree of $S_n$ and $A_n$ over a Field of Characteristi …

For Abelian group $G$, the eigenvalues of the Cayley graph $Cay(G,S)$ can be computed by the group characters and the set $S$. Actually, if $\rho$ be a character of the group $G$, $\rho(S)=\Sigma_{s\i …

Let $G$ be a finite group. We say a subset $A$ of $G$, $|A|=m$, is $(m,i)$-good, $m\geq 1$ and $0\leq i\leq m$, if there exist $g_A\in G$ such that we have $|gA\cap A|=m-i$. I need some groups such t …

I asked some questions about finite Cayley graphs with special type of subgraphs which has been answered by Dear Prof. Godsil. It can be seen in the MO page with address: Cayley graphs and its subgra …

I want to give a lecture about Frucht theorem (and its generalization) which state that: for each finite group $G$, there is at least one finite graph $\Gamma$ such that $Aut(\Gamma)\cong G$. For each …

I have two questions about Cayley graphs. Any answers will be appreciate. 1) Do we have any Cayley graph that has Petersen graph as its induced subgraph? 2) Suppose $Cay(G,S)$ be a Cayley graph that …

Recently Professor Peter Cameron posed a number theory problem which is related to graphs of groups. The problem is: Problem: Let $n$ be a positive integer. Show that there exist subsets $A_1, A_2, … …

Just for other reference, we showed that a finite DS group is solvable, and every non-cyclic Sylow subgroup of a finite DS group is of order $4$, $8$, $16$ or $9$. You can find more results in the bel …

Let $D_{n}$ denotes the dihedral group of order $2n$. Firstly, for self-referencing of the question, I give some definitions which are standard. Let $G$ be a finite group. A subset $S$ of group $G$ i …

Frucht's theorem is a theorem in algebraic graph theory conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite …