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Let L: =U 3 (11). In this article, we classify groups with the same order and degree pattern as an almost simple group related toL. In fact, we prove that L, L: 2 and L: 3 are OD-characterizable, and L: S3 is 5-fold OD-characterizable.

Let G be a finite group and p (G) be the set of all the prime divisors of |G|. The prime graph of G is a simple graph G (G) whose vertex set is p (G) and two distinct vertices p and q are joined by an edge if and only if G has an element of order pq, and in this case we will write p~q. The degree of p is the number of vertices adjacent to p and is denoted by deg (p). If |G|=p1a1p2a2 … pkak, pi’s different primes, p1<p2<…< pk, then D (G) = (deg (p1), deg (p2), …, deg (pk)) is called the degree pattern of G. A finite group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups S with |G|=|S| and D (G) =D (S). In this paper, we characterize groups with the same order and degree pattern as an almost simple groups related to L3 (25).

Let G be a finite group and pe (G) be the set of orders of all elements in G. The set pe (G) determines the prime graph (or GrunbergKegel graph) G (G) whose vertex set is pe (G). The set of primes dividing the order of G, and two vertices p and q are adjacent if and only if pq  Î pe (G). The degree deg (p) of a vertex p Îp (G), is the number of edges incident on p. Let p (G) = {p1, p2,…, pk} with p1 < p2 <…< pk. We define D(G) := (deg (p1), deg (p2), …, deg (pk)), which is called the degree pattern of G. The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups M satisfying conditions |G| = |M| and D(G) = D(M). Usually a 1-fold OD-characterizable group is simply called OD-characterizable. In this paper, we classify all finite groups with the same order and degree pattern as an almost simple groups related to D4 (4).

Let L = U3(9) be the simple projective unitary group in dimension 3 over a field with 92 elements. In this article, we classify groups with the same order and degree pattern as an almost simple group related to L. Since Aut(L)@Z4 hence almost simple groups related to L are L, L : 2 or L : 4. In fact, we prove that L, L : 2 and L : 4 are OD-characterizable.

Let G be a finite group and p(G) be the set of all prime divisors of |G|. The prime graph of G is a simple graph G (G) with vertex set p(G) and two distinct vertices p and q in p(G) are adjacent by an edge if an only if G has an element of order pq. In this case, we write p ~ q. Let |G|=p a11. p a22…p akk, where p1<p2<…<pk are primes. For p Îp(G), let deg (p)=|{qÎp(G) |p ~ q}| be the degree of p in the graph G(G), we define D(G)=(deg (p1), deg (p2),…, deg (pk)) and call it the degree pattern of G. A group G is called k -fold OD-characterizable if there exist exactly k non-isomorphic groups S such that |G|=|S| and D(G)=D(S). Moreover, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Let L=S4 (4) be the projective symplectic group in dimension 4 over a field with 4 elements. In this article, we classify groups with the same order and degree pattern as an almost simple group related to L. Since Aut (L)@Z4 hence almost simple groups related toL are L, L: 2 or L: 4. In fact, we prove that L, L: 2 and L: 4 are OD-characterizable.