We introduce the notion of a powerful action of a p-group upon another p-group. This represents a generalization of powerful p-groups introduced by Lubotzky and Mann in 1987. We derive some properties of powerful actions and study faithful powerful actions. We also prove that the non-abelian tensor product of powerful p-groups acting powerfully and compatibly upon each other is again a powerful p-group.
In this note we describe the exponent semigroups of finite p-groups of maximal class and finite p-groups of class at most 5. Consequently, sharp bounds for the exponent of the Schur multiplier of a finite p-group of class at most 4 are obtained. Our results extend some well-known results of Jones (1974).
A group G is said to be n-central if the factor group G/Z(G) is of exponent n. We improve a result of Gupta and Rhemtulla by showing that every 4-central group is 16-abelian and every 6-central group is 36-abelian. There are examples of finite groups which show that these bounds are best possible. Consequently, we can completely describe the structure of exponent semigroups of free non-cyclic n-central groups for n = 2, 3, 4, 6. We obtain a characterization of metabelian p-central groups and a classification of finitely generated 2-central groups. We compute the nilpotency class of the free metabelian 4-central group of arbitrary finite rank.
We prove that if G is a finite group, then the exponent of its Bogomolov multiplier divides the exponent of G in the following four cases: (i) G is metabelian, (ii) , (iii) G is nilpotent of class , or (iv) G is a 4-Engel group.