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:
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 nonidentity group element. The conjugate closure of the empty set 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.)
References
 Derek F. Holt; Bettina Eick, Eamonn A. O'Brien (2005). Handbook of Computational Group Theory. CRC Press. pp. 73. ISBN 1584883723.
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