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

S^G = {x ∈ G | there exists g ∈ G and s ∈ S such that x = g⁻¹sg},

The conjugate closure of S is denoted <S^G> or <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. For this reason, the conjugate closure is also called the normal closure of S or the normal subgroup generated by S. The normal closure can also be characterized as the intersection of all normal subgroups of G which contain S. If S is already normal then it is equal to its normal closure.

If S = ∅, then the normal closure of S is the trivial group. 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.

Contrast the normal closure of S with the normalizer of S, which is the largest subgroup of G in which <S> is normal.