594: Conjugate Subgroup of Subgroup by Element

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definition of conjugate subgroup of subgroup by element

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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: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of group.

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Target Context

• The reader will have a definition of conjugate subgroup of subgroup by element.

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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:
$G^{\prime }$: $\in \{\text{ the groups }\}$
$G$: $\in \{\text{ the subgroups of }{G}^{\prime }\}$
$p$: $\in G^{\prime }$
$\ast pG{p}^{-1}$: $=\text{ the conjugate subgroup of }G\text{ by }p$
//

Conditions:
//

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2: Natural Language Description

For any group, $G^{\prime }$, any subgroup, $G$, of $G^{\prime }$, and any $p\in G^{\prime }$, $pG{p}^{-1}$ is the conjugate subgroup of $G$ by $p$

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3: Note

$pG{p}^{-1}$ is indeed a subgroup: $1=p1p^{-1}\in pG{p}^{-1}$; for each $p{p}_1{p}^{-1},p{p}_2{p}^{-1}\in pG{p}^{-1}$, $p{p}_1{p}^{-1}p{p}_2{p}^{-1}=p{p}_1{p}_2{p}^{-1}\in pG{p}^{-1}$; for each $p{p}_1{p}^{-1}\in pG{p}^{-1}$, $p{p_1}^{-1}{p}^{-1}\in pG{p}^{-1}$ and $p{p_1}^{-1}{p}^{-1}p{p}_1{p}^{-1}=1=p{p}_1{p}^{-1}p{p_1}^{-1}{p}^{-1}$; associativity obviously holds.

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References

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