Groups Of Prime Exponent Research Articles

Large Abelian Subgroups of Groups of Prime Exponent

Abelian subgroups, particularly ‘‘large’’ abelian subgroups of p-groups, have played an important role in finite group theory. Let p be a prime and let P be a finite p-group. Suppose A is an elementary abelian subgroup of some order p in P. We ask whether P has a normal elementary abelian subgroup of order p. In general, the answer is no, for example, if n 2 and P is the dihedral group of order 16. More generally, Alperin has constructed a counterexample for each prime p Hup, p. 349 . However, there are situations in which the answer must be yes, e.g., AG, Theorems A and 5.8 if p is odd and p 4n 7, if P has nilpotence class at most p, Ž p . or if p 3 and P has exponent p x 1 for every x P . Since Alperin’s counterexamples have exponent greater than p, this leaves the question open if p 5 and P has exponent p. The purpose of this paper is to give counterexamples in these cases. Ž . As usual, define the rank m A of a finite abelian group to be the Ž . minimal number of generators of A, and the rank r P of a finite p-group P to be the maximum of the ranks of the abelian subgroups of P. We prove:

The Restricted Burnside Problem

Preface Contents Notation 1. Basic concepts 2. The associated Lie ring of a group 3. Kostrikin's Theorem 4. Razmyslov's theorem 5. Groups of exponent two, three, and six 6. Groups of exponent four 7. Groups of prime exponent 8. Groups of prime-power exponent 9. Zelmanov's Theorem Appendix A Appendix B Index

Center-By-Metabelian Groups of Prime Exponent

We show that a center-by-metabelian group of prime exponent

Center-by-metabelian groups of prime exponent

We show that a center-by-metabelian group of prime exponent p is nilpotent of class at most p, and this result is best possible. The proof is based on techniques dealing with varieties of groups.

The distribution of sequences in Abelian groups

1Introduction.In this paper, I give a definition of uniform distribution for sequences taking values in certain types of Abelian group—in particular, in groups of prime exponent—and one of independent distribution for sets of such sequences. In the various cases, I find conditions for the properties to hold, which are similar to Weyl's criterion for the uniform distribution of real sequences modulo 1.

Some groups of prime exponent

For each prime p≥5, certain groups of exponent p are exhibited. It follows that the precise solubility length of B 4 p * (the restricted Burnside group of exponent p on 4 generators) is at least [1+log 2( p−1)].