Finite groups with prime 𝑝 to the first power

Abstract

Earlier D. G. Higman classified the finite groups of order n, such that n is divisible by 3 to the first power, with the assumption that the centralizer C G ( X ) {C_G}(X) of X, where X is a subgroup of order 3, is a cyclic trivial intersection set of even order 3s. In this paper the theorem is generalized to include all prime numbers greater than 3. With an additional assumption: | N G ( X ) : C G ( X ) | = 2 |{N_G}(X):{C_G}(X)| = 2 , we have proved that one of the following holds for these groups, hereafter designated as G: (A) G is isomorphic to L 2 ( q ) {L_2}(q) , where q = 2 p s ± 1 q = 2ps \pm 1 ; (B) there exists a normal subgroup G 0 {G_0} of odd index in G, and a normal subgroup N of G 0 {G_0} of index 2 such that G = N ⟨ σ ⟩ G = N\langle \sigma \rangle where C G ( X ) = X × ⟨ σ ⟩ {C_G}(X) = X \times \langle \sigma \rangle .