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A finite group is called a CH-group if for every x, y ∈ G \ Z(G), x y = yx implies that C G (x) = C G (y) . Applying results of Schmidt ['Zentralisatorverbände endlicher Gruppen', Rend. Sem. Mat. Univ. Padova 44 (1970), 97-131] and Rebmann ['F-Gruppen', Arch. Math. 22 (1971), 225-230] concerning CA-groups and F-groups, the structure of CH-groups is determined, up to that of CH-groups of prime-power order. Upper bounds are found for the derived length of nilpotent and solvable CHgroups. 2000doi:10.1017/s0004972710000298 fatcat:lvzlcmsklbfbnm6cvpagucwfzy