Some Sylow subgroups

Abstract

Most of the well-known theorems of Sylow for finite groups and of P. Hall for finite soluble groups have been extended to certain restricted classes of infinite groups. To show the limitations of such generalizations, examples are here constructed of infinite groups subject to stringent but natural restrictions, groups in which certain Sylow or Hall theorems fail. All the groups are metabelian and of exponent 6. There are countable such groups in which a Sylow 2-subgroup has a complement but no Sylow 3-complement; or again no complement at all. There are countable such groups with continuously many mutually non-isomorphic Sylow 3-subgroups. There are groups, necessarily of uncountable order, with Sylow 2-subgroups of different orders. The most elaborate example is of an uncountable group in which all Sylow 2-subgroups and 3-subgroups are countable, and none is complemented.