OD-CHARACTERIZATION OF ALMOST SIMPLE GROUPS RELATED TO U_6(2)

Let G be a finite group and π（G） = {pl,p2,…… ,pk} be the set of the primes dividing the order of G. We define its prime graph F（G） as follows. The vertex set of this graph is 7r（G）, and two distinct vertices p, q are joined by an edge if and only if pq ∈ πe（G）. In this case, we write p - q. For p ∈π（G）, put deg（p） ：= ｜{q ∈ π（G）｜p - q}｜, which is called the degree of p. We also define D（G） ：= （deg（p1）, deg（p2）,..., deg（pk））, where pl 〈 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 l-fold OD-characterizable group is simply called an OD-characterizable group. Let L ：= U6（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.S3 is 5-fold OD-characterizable.

OD-characterization of Almost Simple Groups Related to U_3(5)

Let G be a finite group with order ｜G｜=p1^α1p2^α2……pk^αk, where p1 〈 p2 〈……〈 Pk are prime numbers. One of the well-known simple graphs associated with G is the prime graph （or Gruenberg- Kegel graph） denoted .by г（G） （or GK（G））. This graph is constructed as follows： The vertex set of it is π（G） = {p1,p2,…,pk} and two vertices pi, pj with i≠j are adjacent by an edge （and we write pi - pj） if and only if G contains an element of order pipj. The degree deg（pi） of a vertex pj ∈π（G） is the number of edges incident on pi. We define D（G） ：= （deg（p1）, deg（p2）,..., deg（pk））, which is called the degree pattern of G. A group G is called k-fold OD-characterizable if there exist exactly k non- isomorphic groups H such that ｜H｜ = ｜G｜ and D（H） = D（G）. Moreover, a 1-fold OD-characterizable group is simply called OD-characterizable. Let L ：= U3（5） be the projective special unitary group. In this paper, we classify groups with the same order and degree pattern as an almost simple group related to L. In fact, we obtain that L and L.2 are OD-characterizable; L.3 is 3-fold OD-characterizable; L.S3 is 6-fold OD-characterizable.