On the Engel margin

Tommy Kay Teague

Vol. 50 (1974), No. 1, 205–214

DOI: 10.2140/pjm.1974.50.205

Abstract

The marginal subgroup for any outer commutator word has been characterized by R. F. Turner-Smith. This paper considers the marginal subgroup E(G) of G for the Engel word e₂(x,y) = [x,y,y] of length two. The principal result is that an element a of G is in E(G) if and only if [x,y,a][a,y,x] is a law in G. The method of proof relies upon properties of Engel elements established by W. Kappe.

Among other results are the following: (a) E(G)∕Z₂(G) is an elementary Abelian 3-group of central automorphisms on the commutator subgroup G′. (b) If Z(G) ∩ γ₃(G) has no elements of order 3 or if G′ is Cernikov complete, then E(G) = Z₂(G). (c) If [G : E(G)] = m is finite, then the verbal subgroup e₂(G) is finite with order dividing a power of m.

Mathematical Subject Classification 2000

Primary: 20F10

Milestones

Received: 29 August 1972

Published: 1 January 1974