On the structure of compact torsion groups

Abstract

Abstract. It is shown that every profirdte torsion group has a finite series of closed characteristic subgroups in which each factor either is a pro-p-group for some prime p or is isomorphic to a Cartesian product of isomorphic finite simple groups. w 1. IntroductionIt has been known for some time that all compact Hausdorff torsion groups are profinite. A proof of this result may be found in [3], p. 69. Examples of infinite profinite torsion groups are provided by, for instance, Cartesian products of finite groups of bounded exponent, with the product topology. In [2], HERFORT has shown that if G is a profinite torsion group then the set ~ (G) ofprimesp for which G has non-trivial Sylow p-subgroups is finite. Here we use this result, the results of HALL and HIGMAN [1] and the classification of the finite simple groups to establish a stronger statement: Theorem 1. Let G be a compact Hausdorff torsion group. Then G has a finite series

Conjugacy class conditions in locally compact second countable groups

“…Every torsion pro-p group is locally finite. Theorem 2.6 (Wilson [11]). Let U be a compact Hausdorff torsion group.…”

Nonsoluble and non-p-soluble length of finite groups

“…For example, such bounds were implicitly obtained in the Hall-Higman paper [9] as part of their reduction of the Restricted Burnside Problem to p-groups. Such bound were also a part of Wilson's theorem [16] reducing the problem of local finiteness of periodic profinite groups to pro-p-groups. (Both problems were solved by Zelmanov [17,18,19,20]).…”

On profinite groups with Engel-like conditions

“…In particular, using Wilson's reduction theorem [11], Zelmanov has been able to prove local finiteness of profinite periodic groups [16]. Recall that a group is periodic if all of its elements have finite order.…”