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In [[group theory]], the '''conjugate closure''' of a [[subset]] ''S'' of a [[group (mathematics)|group]] ''G'' is the [[subgroup]] of ''G'' [[generating set of a group | generated]] by ''S''<sup>''G''</sup>, i.e. the closure of ''S''<sup>''G''</sup> under the group operation, where ''S''<sup>''G''</sup> is the [[Conjugate (group theory)|conjugates]] of the elements of ''S'':
:''S''<sup>''G''</sup> = {''g''<sup>&minus;1</sup>''sg'' | ''g'' &isin; ''G'' and ''s'' &isin; ''S''}
The conjugate closure of ''S'' is denoted <''S''<sup>''G''</sup>> or <''S''><sup>''G''</sup>.
 
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 (set theory)|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 set|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 non-identity group element. The conjugate closure of the empty set <math>\varnothing</math> 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==
* {{cite book | title=Handbook of Computational Group Theory | author=Derek F. Holt | coauthors=Bettina Eick, Eamonn A. O'Brien | publisher=CRC Press | year=2005 | isbn=1-58488-372-3 | pages=73 }}
 
[[Category:Group theory]]
 
 
{{Abstract-algebra-stub}}

Revision as of 18:11, 5 January 2014

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 conjugates of the elements of S:

SG = {g−1sg | g ∈ G and s ∈ S}

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, any simple group is the conjugate closure of any non-identity 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

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