Testing Isomorphism of Central Cayley Graphs Over Almost Simple Groups in Polynomial Time

Abstract

A Cayley graph over a group G is said to be central if its connection set is a normal subset of G. It is proved that for any two central Cayley graphs over explicitly given almost simple groups of order n, the set of all isomorphisms from the first graph onto the second can be found in time poly(n).

On Cayley representations of central Cayley graphs over almost simple groups

“…It can be deduced form the classification of finite simple groups (CFSG), see, e.g, Tables 5,6 in p. xvi [4,Introduction], or [11], that The next lemma essentially proved in [18] exploits the classification of regular almost simple subgroups of a primitive group [13,Theorem 1.4].…”

“…Let G be a finite group and [22,Theorem 24.12]); moreover, X is normal in G because K ≥ G l . Denote by L = L(K) the intersection of all non-singleton blocks of K containing e. Following [18], we say that L is the minimal block of K. Denote the block system of K containing L by L = L(K). Clearly, L coincides with the set of all L-cosets, L * ≤ K L , and Orb(L * , G) = L.…”

“…Following [18], we say that K is of normal type provided Item (i) of Lemma 5.2 holds, and of symmetric type otherwise.…”

“…We begin with the observation that a generating set of K can be found in time poly(n) by [18,Corollary 1.2]. Applying standard algorithms for permutation groups (here our main source is [20]), one can find the socle S r = Soc(G r ), the minimal block L = L(K) and L = L(K) in time poly(n), and check whether K is of symmetric type or normal type.…”

“…Some general description of the full automorphism group of a connected Cayley graph over a simple group was given in [8]. A polynomial algorithm solving the isomorphism problem for central Cayley graphs over almost simple groups was proposed in [18].…”