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normal closure (Definition)

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

$\displaystyle \bigcap_{S\subseteq N\trianglelefteq G}\!\!N.$
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.



"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 7 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 5261 times total.

Classification:
AMS MSC: 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)
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