Finite p-Group with Small Abelian Subgroups

Abstract EN

A finite p-group G is said to have the property P, if, for any abelian subgroup M of G, there is |MZ(G)/Z(G)|≤p.

We show that if G satisfies P, then G has the following two types: (1) G is isoclinic to some stem groups of order p5, which form an isoclinic family.

(2) G is isoclinic to a special p-group of exponent p.

Elementary structures of groups with P are determined.