Abstract

A proper subgroup H of a group G is called a CC-subgroup of G if the centralizer CG(h) of h ∈ H# = H \ {1} is contained in H. Such finite groups were partially classified by G. Frobenius, W. Feit, K. W. Gruenberg and O. H. Kegel, J.S. Williams, A. S.Kondrat'iev, N. Iiyori and H.Yamaki, M. Suzuki, M. Herzog, Z. Arad, D. Chillag, Ch. Praeger and others.

In this report, using the classification of finite simple groups, we give a complete list of all finite groups containing a CC-subgroup. As a corollary we classify infinite profinite groups, locally finite groups and certain classes of topological groups containing a CC-subgroup under certain conditions.

Introduction

Let G denote a finite group. According to M. Herzog [18] a subgroup M ≤ G is a CC-subgroup (”centralizers contained“), if CG(m) ≤ M for every m ∈ M \ {1}. The example with smallest cardinality is G := S3 with either M := 〈(123)〉 or M := 〈(12)〉 being a CC-subgroup. More generally, by the well known result of G. Frobenius, every Frobenius group has CC-subgroups either the kernel or any complement.

Sketching the thread

One finds the concept of a CC-subgroup (without calling it that) in work of W. Feit describing doubly transitive groups which fix 3 letters (e.g. in [13]).