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 Alternating and Symmetric Groups of Degree p+5

In this paper,it is proved that all the alternating groups A_(p+5) are ODcharacterizable and the symmetric groups S_(p+5) are 3-fold OD-characterizable,where p + 4 is a composite number and p + 6 is a prime and 5≠p∈π(1000!).

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.

Recognizing Finite Groups Through Order and Degree Patterns

The degree pattern of a finite group G associated with its prime graph has been introduced by Moghaddamfar in 2005 and it is proved that the following simple groups are uniquely determined by their order and degree patterns： All sporadic simple groups, the alternating groups Ap （p ≤ 5 is a twin prime） and some simple groups of the Lie type. In this paper, the authors continue this investigation. In particular, the authors show that the symmetric groups Sp＋3, where p ＋ 2 is a composite number and p ＋ 4 is a prime and 97 〈 p ∈π（1000!）, are 3-fold OD-characterizable. The authors also show that the alternating groups All6 and A134 are OD-characterizable. It is worth mentioning that the latter not only generalizes the results by Hoseini in 2010 but also gives a positive answer to a conjecture by Moghaddamfar in 2009.