Nilpotency in groups with the minimal condition on centralizers
The paper makes a number of substantial contributions to the theory of groups with the minimal condition on centralizers, called \({\mathfrak M}_c\)-groups: these are the groups in which there is no infinite descending (or, equivalently, ascending) chain of subgroups each of which is the centralizer of some subset of the group. Three of the main results are as follows. (1) Suppose that \(G\) is an \({\mathfrak M}_c\)-group which is also a \(p\)-group for some prime \(p\): then \(G\) is locally finite if either \(p=2\) or every pair of elements of \(G\) generates a finite subgroup. The case \(p=2\) answers, positively, a question of John Wilson. (2) Suppose that \(G\) is a periodic \({\mathfrak M}_c\)-group: then the Sylow \(p\)-subgroups of \(G\) are conjugate if either \(p=2\) or every pair of \(p\)-elements of \(G\) generates a finite subgroup. This generalizes a result of the reviewer for locally finite \({\mathfrak M}_c\)-groups [J. Algebra 60, 371-383 (1979; Zbl 0422.20022)]. (3) In any \({\mathfrak M}_c\)-group the bounded left Engel elements form the Fitting subgroup. (An element \(g\) of a group \(G\) is bounded left Engel if there exists a positive integer \(n\) such that \([{x,_n}g]=1\) for all \(x\) in \(G\).) This generalizes a similar result for `substable' groups (see the paper under review for references to this and other related earlier work).
scientific article; zbMATH DE number 1340556