Conjugate closure

In group theory, the conjugate closure of a subset S of a group G is the subgroup of G which is generated by the elements of S and their conjugates. Writing S^G for the set {x in G: exists g in G and s in S such that x = g ^-1sg}, we sometimes notate the conjugate closure of S as <S^G>.

The conjugate closure of S is always a normal subgroup of G; in fact, it is the smallest (by inclusion) normal subgroup of G which contains <S>, the subgroup generated by the elements of S. We can compare this to the normalizer of S, which is the largest subgroup of G in which <S> is normal.

If S = {a} consists of a single element, then the conjugate closure is a normal subgroup generated by a and all elements of G which are conjugate to a. Therefore, if G is a simple group, G is generated by the conjugate closure of any non-identity element a of G.