A note on the continuity of the inverse

Abstract

In his article [2] Wallace mentions the following problem: let X be an algebraic group with a locally compact Hausdorff topology such that the map of XXX into X which takes (x, y) into xy for all x, yEX is continuous. Then is X a topological group? The purpose of this note is to answer this, question in the affirmative. Lemma 1 is an immediate consequence of the continuity of multiplication, and the proof of Lemma 2 appears in [1]. The proofs of these lemmas will therefore be omitted.