Conjugate closure

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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:

SG = {g−1sg | gG and sS}

The conjugate closure of S is denoted <SG> 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[clarification needed], any simple group is the conjugate closure of any non-identity group element. The conjugate closure of the empty set \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]

References[edit]

  1. ^ Robinson p.16