arXiv subjects
Publications and source records attributed to Paul J. Sanders.
For a prime p and natural number n with p greater than or equal to n, we establish the existence of a non-functorial one-to-one correspondence between isomorphism classes of groups of order p^n whose derived subgroup has exponent dividing p, and isomorphism classes of nilpotent p^n-element Lie algebras L over the truncated polynomial ring F_p[T]/(T^n) in which T[L,L]=0.
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 p^n 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.
The coexponent of a finite p-group is introduced and we consider how the nilpotency class is bounded in terms of this invariant.
It is shown how to use a small finite state automaton in two variables in order to carry out part of the Knuth--Bendix process for rewriting words in a group. The main objective is to provide a substitute for the most space-demanding module of the existing software which attempts to find a shortlex-automatic structure for a group. The two-variable automaton can be used to store an infinite set of rules and to carry out fast reduction of arbitrary words using this infinite set. We introduce a new operation, which we call welding, which applies to an arbitrary finite state automaton. In our context this operation is vital. We point out a small potential improvement in the subset algorithm for making a non-deterministic automaton deterministic.