Some Quantitative Characterizations of Certain Symplectic Groups

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Given a finite group G, denote by D(G) the degree pattern of G and by OC(G) the set of all order components of G. Denote by h OD(G) (resp. h OC(G)) the number of isomorphism classes of finite groups H satisfying conditions |H|=|G| and D(H)= D(G) (resp. OC(H)= OC(G)). A finite group G is called OD-characterizable (resp. OC-characterizable) if h OD(G)=1 (resp. h OC(G)=1). Let C=Cₚ(2) be a symplectic group over binary field, for which 2ᵖ-1>7 is a Mersenne prime. The aim of this article is to prove that h OD(C)=1=h OC(C).