Conjugate closure

From Academic Kids

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

SG = {xG | there exists gG and sS such that x = g−1sg},

The conjugate closure of S is denoted <SG> 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 generated by 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> is normal.

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