713: Centralizer of Element on Group

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definition of centralizer of element 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: 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 centralizer of element 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:
$G$: $\in \{\text{ the groups }\}$
$p$: $\in G$
$\ast C_G(p)$: $=\{p^{\prime }\in G|p^{\prime }p{p}^{\prime -1}=p\}$, $\in \{\text{ the subgroups of }G\}$
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Conditions:
//

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2: Natural Language Description

For any group, $G$, and any element, $p\in G$, the subgroup of $G$, $C_G(p):=\{p^{\prime }\in G|p^{\prime }p{p}^{\prime -1}=p\}$

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3: Note

$C_G(p)$ is indeed a subgroup: 0) for each $p^{\prime },p^{″}\in C_G(p)$, $p^{\prime }{p}^{″}p(p^{\prime }{p}^{″}{)}^{-1}=p^{\prime }{p}^{″}p{p^{″}}^{-1}{p}^{\prime -1}=p^{\prime }p{p}^{\prime -1}=p$; 1) for any elements, $p_1,p_2,p_3\in C_G(p)$, $(p_1{p}_2)p_3=p_1(p_2{p}_3)$, because it holds in the ambient $G$; 2) the identity element is in $C_G(p)$, because $1p{1}^{-1}=p$; 3) for each element, the inverse is in $C_G(p)$, because $p^{\prime -1}p(p^{\prime -1}{)}^{-1}=p^{\prime -1}p{p}^{\prime }=p^{\prime -1}{p}^{\prime }p{p}^{\prime -1}{p}^{\prime }=p$.

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References

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