It is well known that the set of values a lower central word in group G need not be subgroup. For fixed ?r and for p ? 5, Guralnick showed if finite p-group such verbal subgroup ?r(G) abelian 2-generator, then consists only ?r-values. In this paper we extend result, showing assumption can dropped. Moreover, show result remains true even p= 3. Finally, prove analogous pro-p groups true.

DEFINITION. A group G is called a universal locally finite central extension of A provided that the following conditions are satisfied. (i) A <= (G (the centre of G). (ii) G is locally finite. (iii) (/1-injectivity). Suppose that A <= B <= D with A a (D, that D/A is finite, and that q>: B -> G is an ^-isomorphism (that is, q>{a) = a for all as A). Then there exists an extension q>: D -*• G of (...

The purpose of this note is to describe how central multiplier theorems for compact Lie groups can be reduced to corresponding results on a maximal torus. We shall show that every multiplier theorem for multiple Fourier series gives rise to a corresponding theorem for such groups and, also, for expansions in terms of special functions. We use the notation and terminology of N. J. Weiss [4]. Let...

A general method for calculating or constructing lower central factors of groups is presented. Relative basic commutators are defined.

The so-called Burnside-Dixon-Schneider (BDS) method currently used as the default method of computing character tables in GAP for groups which are not solvable is often inefficient in dealing with groups with large centres. If G is a finite group with centre Z and λ a linear character of Z, then we describe a method of computing the set Irr(G,λ) of irreducible characters χ of G whose restrictio...

An extension of a group A by a group G is thought of here simply as a group H containing A as a normal subgroup with quotient H/A isomorphic to G. It is called a central Frattini extension if A is contained in the intersection of the centre and the Frattini subgroup of H . The result of the paper is that, given a finite abelian A and finite G, there exists a central Frattini extension of A by G...