The Maier–Schmid problem for infinite groups

The Maier–Schmid theorem asserts that a core-free permutable subgroup of a finite group is contained in the hypercentre. This result is known to be false even for finitely generated groups. Here it is shown that the Maier–Schmid theorem holds for many classes of infinite groups, including numerous locally finite groups. On the other hand, an example is given of a countable metabelian p-group for which the theorem fails.