Automorphism groups of tetravalent Cayley graphs on regular [formula omitted]-groups

The automorphism groups of Cayley graphs on symmetric groups, Cay(G, S), where S is a complete set of transpositions have been determined. In a similar spirit, automorphism groups of Cayley graphs Cay(An , S) on alternating groups An , where S is a set of all 3-cycles have also been determined. It has, in addition, been shown that these graphs are not normal. In all these Cayley graphs, one observes that their corresponding Cayley sets are a union of conjugacy classes. In this paper, we determine in their generality, the automorphism groups of Cay(G, S), where G ∈ {An , Sn } and S is a conjugacy class type Cayley set. Further, we show that the family of these graphs form a Boolean algebra. It is first shown that Aut(Cay(G, S)), S ∉ {∅, G \\ {e}}, is primitive if and only if G = An . Using one of the results obtained by Praeger in 1990, we exploit further the other cases, thereby proving that, for n > 4 and n ≠ 6, Aut(Cay(An , S)) ≅ Hol(An ) ⋊ 2, with Hol(G) ∼= G ⋊ Aut(G), provided that S is preserved by the outer automorphism defined by the conjugation by an odd permutation. Finally, in the remaining case G = Sn , n > 4 and n ≠ 6, we show that Aut(Cay(Sn, S) ≅ (Hol(An ) ⋊ 2) ≀ S 2 for S ⊂ An \\ {e}, and that Aut(Cay(Sn , S)) ≅ Hol(Sn ) ⋊ 2 otherwise; provided that S does not contain Sn \\ An or S ≠ An \\ {e}, S ∉ {∅, Sn \\ {e}}.

Automorphism groups of Cayley graphs generated by connected transposition sets

The automorphism groups Aut(C(G, X)) and Aut(CM(G, X, p)) of a Cayley graph C(G, X) and a Cayley map CM(G, X, p) both contain an isomorphic copy of the underlying group G acting via left translations. In our paper, we show that both automorphism groups are rotary extensions of the group G by the stabilizer subgroup of the vertex 1 G . We use this description to derive necessary and sufficient conditions to be satisfied by a finite group in order to be the (full) automorphism group of a Cayley graph or map and classify all the finite groups that can be represented as the (full) automorphism group of some Cayley graph or map.

Automorphism groups of Cayley graphs on symmetric groups with generating transposition sets

A Cayley graph $${\varGamma }=\mathsf{Cay}(G,S)$$ is said to be normal if G is normal in $$\mathsf{Aut}{\varGamma }$$ . The concept of normal Cayley graphs was first proposed by Xu (Discrete Math 182:309–319, 1998) and it plays an important role in determining the full automorphism groups of Cayley graphs. In this paper, we study the normality of connected arc-transitive pentavalent Cayley graphs $${\varGamma }$$ on finite nonabelian simple groups G, where the vertex stabilizer $$\mathsf{A}_v$$ is soluble for $$\mathsf{A}=\mathsf{Aut}{\varGamma }$$ and $$v\in V{\varGamma }$$ . We prove that $${\varGamma }$$ is either normal or $$G=\mathsf{A}_{39}$$ or $$\mathsf{A}_{79}$$ . Further, a connected pentavalent arc-transitive non-normal Cayley graph on $$\mathsf{A}_{79}$$ is constructed. To our knowledge, this is the first known example of pentavalent 3-arc-transitive Cayley graph on finite nonabelian simple group which is non-normal.

Let G be a nite nonabelian simple group and let be a connected undirected Cayley graph for G. The possible structures for the full automorphism group Aut are specied. Then, for certain nite simple groups G, a sucient condition is given under which G is a normal subgroup of Aut. Finally, as an application of these results, several new half-transitive graphs are constructed. Some of these involve the sporadic simple groups G =J 1 ,J 4, Ly and BM, while others fall into two innite families and involve the Ree simple groups and alternating groups. The two innite families contain examples of half-transitive graphs of arbitrarily large valency.

In this paper we study the Cayley graph $Cay(S_n,T)$ of the symmetric group $S_n$ generated by a set of transpositions $T$. We show that for $n\geq 5$ the Cayley graph is normal. As a corollary, we show that its automorphism group is a direct product of $S_n$ and the automorphism group of the transposition graph associated to $T$. This provides an affirmative answer to a conjecture raised by A. Ganesan, Cayley graphs and symmetric interconnection networks, showing that $Cay(S_n,T)$ is normal if and only if the transposition graph is not $C_4$ or $K_n$.

Automorphism groups of Cayley graphs generated by block transpositions and regular Cayley maps

Variants of some of the Brauer-Fowler theorems

A Cayley graph Γ=Cay(G,S) is said to be normal if the base group G is normal in AutΓ. The concept of the normality of Cayley graphs was first proposed by M.Y. Xu in 1998 and it plays a vital role in determining the full automorphism groups of Cayley graphs. In this paper, we construct an example of a 2-arc transitive hexavalent nonnormal Cayley graph on the alternating group A119. Furthermore, we determine the full automorphism group of this graph and show that it is isomorphic to A120.

LetXG,H denote the Cayley graph of a finite groupG with respect to a subsetH. It is well-known that its automorphism groupA(XG,H) must contain the regular subgroupLG corresponding to the set of left multiplications by elements ofG. This paper is concerned with minimizing the index [A(XG,H)∶LG] for givenG, in particular when this index is always greater than 1. IfG is abelian but not one of seven exceptional groups, then a Cayley graph ofG exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.

On cubic Cayley graphs of finite simple groups

This paper inverstigates the automorphism groups of Cayley graphs of metracyclicp-gorups. A characterization is given of the automorphism groups of Cayley grahs of a metacyclicp-group for odd primep. In particular, a complete determiniation of the automophism group of a connected Cayley graph with valency less than 2pof a nonabelian metacyclicp-group is obtained as a consequence. In subsequent work, the result of this paper has been applied to solve several problems in graph theory.

The present thesis embraces two major areas of mathematics, namely group theory (especially growth in finite groups) and graph theory (especially the graph isomorphism problem). Several results are presented coming from both areas: on one side, we show that the dependence of the diameter of a product of finite simple groups on the diameter of its factors is linear, and we extend the analysis of sets of small growth in the affine group over a prime field to the same group over general finite fields; on the other, we show a dependence of the number of iterations of the Weisfeiler-Leman algorithm over Schreier and Cayley graphs on the diameter of such graphs. Finally, analyzing Babai's algorithm for solving the graph isomorphism problem, we pave a possible way towards a proof of a diameter bound for the alternating group that does not rely on the classification of finite simple groups.

This paper is a contribution to the ongoing project of Gorenstein, Lyons, and Solomon to produce a complete unified proof of the classification of finite simple groups. A part of this project deals with classification and characterization of bicharacteristic finite simple groups. This paper contributes to that particular situation.