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normal closure
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(Definition)
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Let $S$ be a subset of a group $G$ The normal closure of $S$ in $G$ is the intersection of all normal subgroups of $G$ that contain $S$ that is $$\bigcap_{S\subseteq N\normal G}\!\!N.$$ The normal closure of $S$ is the smallest normal subgroup of
$G$ that contains $S$ and so is also called the normal subgroup generated by $S$
It is not difficult to show that the normal closure of $S$ is the subgroup generated by all the conjugates of elements of $S$
The normal closure of $S$ in $G$ is variously denoted by $\langle S^G\rangle$ or $\langle S\rangle^G$ or $S^G$
If $H$ is a subgroup of $G$ and $H$ is of finite index in its normal closure, then $H$ is said to be nearly normal.
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"normal closure" is owned by yark.
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See Also: normalizer, core of a subgroup
| Other names: |
normal subgroup generated by, conjugate closure, smallest normal subgroup containing |
| Also defines: |
nearly normal |
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Cross-references: index, finite, subgroup, conjugates, subgroup generated by, contain, normal subgroups, intersection, group, subset
There are 6 references to this entry.
This is version 6 of normal closure, born on 2004-10-06, modified 2006-03-02.
Object id is 6307, canonical name is NormalClosure2.
Accessed 7091 times total.
Classification:
| AMS MSC: | 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties) |
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Pending Errata and Addenda
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