OD-characterization of Almost Simple Groups Related to U3（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.

OD-characterization of almost simple groups related to U 3(5)

Let G be a finite group with order |G| = p ₁ ^α1 p ₂ ^α2 … p _k ^αk , where p ₁ < p ₂ < … < p _k are prime numbers. One of the well-known simple graphs associated with G is the prime graph (or Gruenberg-Kegel graph) denoted by Γ(G) (or GK(G)). This graph is constructed as follows: The vertex set of it is π(G) = {p ₁, p ₂, …, p _k } and two vertices p _i , p _j with i ≠ j are adjacent by an edge (and we write p _i ∼ p _j ) if and only if G contains an element of order p _i p _j . The degree deg(p _i ) of a vertex p _i ∈ π(G) is the number of edges incident on p _i . We define D(G):= (deg(p ₁), deg(p ₂), …, deg(p _k )), which is called the degree pattern of G. A group G is called k-fold OD-characterizable if there exist exactly k nonisomorphic 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:= U ₃(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.S ₃ is 6-fold OD-characterizable.