A sharp bound is derived for the nilpotency class of a regular p-group in terms of its coexponent, and is used to show that the number of groups of order pn with a given fixed coexponent, is independent of n, for p and n sufficiently large. Explicit formulae are calculated in the case of coexponent 3.

In 1956, Green provided a bound on the order of Schur multiplier p-groups. This bound, given as function group, is best possible. Since then, has been refined numerous times by adding other inputs to function, such as, minimal number generators group and derived subgroup. We strengthen these bounds another input, group's nilpotency class. The specific cases class 2 maximal are discussed in grea...

We consider logical deenability of the group-theoretic notions of simplicity, nilpotency and solvability on the class of nite groups. On one hand, we show that these notions are deenable by sentences of deterministic transitive closure logic DTC. These results are based on known group-theoretic results. On the other hand, we prove that simplicity, nilpotency and the normal closure of a subset o...

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

We prove nilpotency results for Lie algebras over an arbitrary field admitting a derivation, which satisfies given polynomial identity r(t) = 0. In the special case of r=tn−1 we obtain uniform bound on class periodic derivation order n. even find optimal in characteristic p if does not divide certain invariant ρn.

The semigroup of the homotopy classes of the self-homotopy maps of a finite complex which induce the trivial homomorphism on homotopy groups is nilpotent. We determine the nilpotency of these semigroups of compact Lie groups and finite Hopf spaces of rank 2. We also study the nilpotency of semigroups for Lie groups of higher rank. Especially, we give Lie groups with the nilpotency of the semigr...