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Z. Arad, W. Herfort:
"Topologically Locally Finite Groups with a CC-Subgroup";
Journal of Lie Theory, 15 (2005), (5); S. 235 - 248.

Classification of locally compact totally disconnected topologically locally finite groups (every compact subset is contained in a compact subgroup), which contain a CC-subgroup, i.e., a proper subgroup, which contains with every nontrivial element its centralizer, and such that a local system of compact subgroups exists, such that M intersected with each member of the system is a Hall subgroup.

http://aleph.ub.tuwien.ac.at/F?base=tuw01&func=find-c&ccl_term=AC05937602