Conjugate closure


In group theory, the conjugate closure of a subset S of a group G is the subgroup of G generated by SG, i.e. the closure of SG under the group operation, where SG is the set of the conjugates of the elements of S:
The conjugate closure of S is denoted <SG> or <S>G.
The conjugate closure of any non-empty subset S of a group G is always a normal subgroup of G; in fact, it is the smallest normal subgroup of G which contains the non-empty set S. For this reason, the conjugate closure coincides with the normal closure of S or the normal subgroup generated by S for non-empty S, where the normal closure is defined 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 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. Note that the conjugate closure of the empty set is empty, while its normal closure is the trivial group.
Contrast the normal closure of S with the normalizer of S, which is the largest subgroup of G in which S itself is normal.
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.