On finite groups whose 2-Sylow subgroups have cyclic subgroups of index 2

Abstract

If the finite group G has a 2-Sylow subgroup S of order 2a+1, containing a cyclic subgroup of index 2, then in general S may be one of the following six types [8]:(i) cyclic; (ii) Abelian of type (a, 1), a > 1; (iii) dihedral1; (iv) generalized quaternion; (v) {α, β}, α2a = β2, α2a−1+1, a ≧ 3;(vi) {α, β}, α2a = β2, α2a−1+1, a ≧ 3.