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 nonidentity element a of G.
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