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Let GT be an equational axiomatization of group theory in terms of a binary multiplication, unary inverse, and a nullary identity element. Let CS be the equational theory with one binary multiplication which is associative and cancellative. Let A be a set of equations (strictly, identities) in the language of one binary operation and let a=b be an equation of the same type.

This is really a meta-conjecture and perhaps is not true in general. However, there are many implications in GT which are provable in "pure" CS (i.e. without the additional luxury of having an inverse or an identity element). Motivated by some special cases of this conjecture, late Professor B.H. Neumann recently published an article on this topic [1] in which he provided many test cases for this conjecture. In this talk, I would like to give an update on this topic and investigate some familiar group-theoretical concepts which can be captured within the language of semigroups.

I thank David Kelly for suggesting this topic for the Seminar.

[1] B.H. Neumann, Some semigroup laws in groups, Canadian Math. Bull. 44 (2001) pp. 93--96.