Conjugate closure

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In group theory, the conjugate closure of a subset S of a group G is the subgroup of G generated by S^G, i.e. the closure of S^G under the group operation, where S^G is the conjugates of the elements of S:

S^G = {g⁻¹sg | g ∈ G and s ∈ S}

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 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> itself is normal. (This need not be normal in the larger group G, just as <S> need not be normal in its conjugate/normal closure.)