Semigroup Forum Vol. 66 (2003) 484–488 c  2002 Springer-Verlag New York Inc. DOI: 10.1007/s002330010150 SHORT NOTE A Note about Congruences on Subsemigroups of Groups Brigitte E. Breckner and Wolfgang A. F. Ruppert Communicated by Jimmie D. Lawson Dedicated to Karl Heinrich Hofmann on the occasion of his 70 th birthday Abstract A well known elementary argument shows that a totally disconnected normal subgroup of a connected topological group is central, in [2] K. H. Hofmann has shown that by a skillful application of this argument (or variations of it) a muchwiderclassofinterestingcentralityresultscanbeobtained. Inthepresent paper we offer a modification of the argument also applying to congruences on subsemigroups of topological groups. Key words and phrases: Congruences on semigroups, centrality of normal subgroups, foliations of congruences 2000 Mathematics Subject Classification: 22A15, 22A05, 22E15, 22E46 1. Centralizers and congruences Proposition 1.1. Let S be a subsemigroup of a topological group [a locally connected topological group] G , and let ρ be a congruence relation on S . Sup- pose that for some a ∈ S there exist interior points u 0 and v 0 of S in G such that u 0 av 0 is an isolated point [ u 0 av 0 has a totally disconnected neighborhood] in its congruence class ρ(u 0 av 0 ) . Then there exists an open subgroup G 1 of G such that the restriction of the inner automorphism x → b -1 xb to G 1 is the same for all b ∈ ρ(a) , or, equivalently, the set ρ(a)a -1 lies in the centralizer of G 1 . Proof. Let O be a neighborhood of u 0 av 0 in G such that O ∩ ρ(u 0 av 0 )= {u 0 av 0 } [such that O ∩ ρ(u 0 av 0 ) is totally disconnected]. Pick b ∈ ρ(a) and let U be an open symmetric [an open symmetric and connected] neighborhood of the identity in G such that (1) Uu 0 ⊆ S , (2) v 0 (u 0 bv 0 ) -1 Uu 0 bv 0 ⊆ S. For every u ∈ U we put v u =(u 0 bv 0 ) -1 u -1 u 0 bv 0 . Note that v u → 1 if u → 1 . Choosing U sufficiently small we enforce that uu 0 av 0 v u ∈ O for each u ∈ U .