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 Group theory is that branch of mathematics concerned with the study of groups. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... Conjugate can be: in mathematics in terms of complex numbers, the complex conjugate; more generally see conjugate element (field theory). ...

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. In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...

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. The following list in mathematics contains the finite groups of small order up to group isomorphism. ... In mathematics, a simple group is a group G such that G is not the trivial group and the only normal subgroups of G are the trivial group and G itself. ...

Contrast the normal closure of S with the normalizer of S, which is the largest subgroup of G in which <S> is normal. In group theory, the centralizer and normalizer of a subset S of a group G are subgroups of G which have a restricted action on the elements of S and S as a whole, respectively. ...