Conjugate closure

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 set of 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 any subset S of a group G 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. Any normal subgroup is equal to its normal closure.

The conjugate closure of a singleton subset {a} of a group G is a normal subgroup generated by a and all elements of G which are conjugate to a. Therefore, any simple group is the conjugate closure of any non-identity group element. The conjugate closure of the empty set ${\displaystyle \varnothing }$ is the trivial group.

Contrast the normal closure of S with the normalizer of S, which is (for S a group) 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.)

Dual to the concept of normal closure is that of normal interior or normal core, defined as the join of all normal subgroups contained in S.[1]