OD-characterization of almost simple groups related to U 6(2)

Abstract

Let G be a finite group and π( G)={p1, p2,…, pk} be the set of the primes dividing the order of G. We define its prime graph γ( G) as follows. The vertex set of this graph is π( G), and two distinct vertices p, q are joined by an edge if and only if pq ɛπ e( G). In this case, we write p∼q. For pɛπ( G), put deg(p):=|{qɛ π(G) | p∼ q|, which is called the degree of p. We also define D( G):=(deg(p1), deg(p2), …, deg(pk)), where p1 < p2 < … < pk, which is called the degree pattern of G. We say a group G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and degree pattern as G. Specially, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Let L:=U 6(2). In this article, we classify all finite groups with the same order and degree pattern as an almost simple groups related to L. In fact, we prove that L and L.2 are OD-characterizable, L.3 is 3-fold OD-characterizable, and L.S 3 is 5-fold OD-characterizable.