871: Normalizer of Subgroup on Group

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definition of normalizer of subgroup on group

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Topics

About: group

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of normal subgroup of group.

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Target Context

• The reader will have a definition of normalizer of subgroup on group.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$G^{\prime }$: $\in \{\text{ the groups }\}$
$G$: $\in \{\text{ the subgroups of }{G}^{\prime }\}$
$\ast N_{G^{\prime }}(G)$: $=\{g^{\prime }\in G^{\prime }|g^{\prime }G{g}^{\prime -1}=G\}$
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Conditions:
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2: Note

The name has originated from the fact that $N_{G^{\prime }}(G)$ is the largest subgroup of $G^{\prime }$ of which ($N_{G^{\prime }}(G)$) $G$ is a normal subgroup.

Let us confirm the fact.

Let us see that $N_{G^{\prime }}(G)$ is a group.

For each $h_1,h_2\in N_{G^{\prime }}(G)$, $h_1{h}_2\in N_{G^{\prime }}(G)$, because $h_1{h}_{2}G(h_1{h}_2{)}^{-1}=h_1{h}_{2}G{h_2}^{-1}{h_1}^{-1}=h_{1}G{h_1}^{-1}=G$.

$1\in N_{G^{\prime }}(G)$, because $1G{1}^{-1}=G$.

For each $h\in N_{G^{\prime }}(G)$, $h^{-1}\in N_{G^{\prime }}(G)$, because $h^{-1}Gh=h^{-1}(hG{h}^{-1})h=(h^{-1}h)G(h^{-1}h)=1G1=G$.

Associativity holds because it holds in the ambient $G^{\prime }$.

$G\subseteq N_{G^{\prime }}(G)$, because for each $g\in G$, $gG{g}^{-1}=G$.

$G$ is a normal subgroup of $N_{G^{\prime }}(G)$, because for each $h\in N_{G^{\prime }}(G)$, $hG{h}^{-1}$.

$N_{G^{\prime }}(G)$ is such the largest, because if there is a subgroup of $G^{\prime }$, $G^{″}$, such that $G$ is a normal subgroup of $G^{″}$, for each $g^{″}\in G^{″}$, $g^{″}G{g}^{″-1}=G$, which means that $g^{″}\in N_{G^{\prime }}(G)$, which means that $G^{″}\subseteq N_{G^{\prime }}(G)$.

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References

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