Topologically locally finite groups with a CC-subgroup

A proper subgroup M of a finite group G is called a CC-subgroup of G if the centralizer C_G(m) of every m ε M^#=M \{1} is contained in M. Such finite groups had been partially classified by S. WILLIAMS, A. S. KONDRAT'IEV, N. IIYORI and H. YAMAKI, M. SUZUKI, W. FEIT and J.G. THOMPSON, M. HERZOG, Z. ARAD, D. CHILLAG and others. In [6] the present authors, having taken all this work into account, classified all finite groups containing a CC-subgroup. As an application, in the present paper, we classify totally disconnected topologically locally finite groups, containing a topological analogue of a CC-subgroup.