We give a classification of two-generator p-groups of nilpotency class 2. Using this classification, we give a formula for the number of such groups of order p in terms of the partitions of n of length 3, and find formulas for the number and size of their conjugacy classes.

In this paper we describe the structure of locally finite groups in which the bounded nilpotency of two-generator subgroups is a transitive relation. We also introduce the notion of (nilpotent of class c)-transitive kernel. Our results generalize several known results related to the groups in which commutativity is a transitive relation. MSC 2000: 20E15, 20D25

Starting from a non-standard definition, the descent algebra of the symmetric group is investigated. Homomorphisms into the tensor product of smaller descent algebras are defined. They are used to construct the irreducible representations and to obtain the nilpotency index of the radical.

In this paper we address the problem of understanding when a verbal subgroup finite group is p -nilpotent, with prime, that is, all its elements ′ -order determine subgroup. We provide two -nilpotency criteria, one for terms lower central series any and derived soluble group, which relies on arithmetic properties related to order products commutators .