Second-metacyclic finite 2-groups

Abstract

Abstract. Second-metacyclic finite 2-groups are finite 2-groups with some non-metacyclic maximal subgroup and with all second-maximal subgroups being metacyclic. According to a known result there are only four non-metacyclic finite 2-groups with all maximal subgroups being metacyclic. The groups pointed in the title should contain some of these groups as a subgroup of index 2. There are seventeen second-maximal finite 2-groups, four among them being of order 16, ten of order 32 and three of order 64.

On finite nonabelian 2-groups all of whose minimal nonabelian subgroups are of exponent 4

“…Hence, C G (A) A and so d(G) = 3, a contradiction. Thus, A is abelian of type (4,2) or (4,4). We have (yx) 2 = y 2 (x y x), where o(y 2 ) 2.…”

“…If [x, x y ] = 1, then A is abelian of type (4,2) or (4,4). If [x, x y ] = 1, then [x, x y ] ∈ x ∩ x y and so [x, x y ] is a central involution in A which implies that A is minimal nonabelian.…”

“…Finally, assume that |K| = 2 6 . Each maximal subgroup of K is minimal nonmetacyclic of order 2 5 and such a group K is unique according to [4,Theorem 2 and Remarks] so K is special with Z(K) ∼ = E 4 . We have obtained the groups stated in parts (b1), (b2) and (b3) of our theorem.…”

“…2 Theorem 2.7. Let G be a nonabelian 2-group all of whose minimal nonabelian subgroups are isomorphic to H 16 = a, t | a4 = t 2 = 1, [a, t] = z, z 2 = [a, z] = [t, z] = 1 . Then Ω 1 (G)is a self-centralizing elementary abelian subgroup and G is of exponent 4.…”