On groups generated by involutions of a semigroup

An involution on a semigroup S (or any algebra with an underlying associative binary operation) is a function α : S → S that satisfies α(xy) = α(y)α(x) and α(α(x)) = x for all x, y ∈ S. The set I(S) of all such involutions on S generates a subgroup C (S) = I(S) of the symmetric group Sym(S) on the set S. We investigate the groups C (S) for certain classes of semigroups S, and also consider the question of which groups are isomorphic to C (S) for a suitable semigroup S.