842: Center of Group

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definition of center of 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 group.

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Target Context

• The reader will have a definition of center of 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:
$G$: $\in \{\text{ the groups }\}$
$\ast Z(G)$: $=\{g\in G|\mathrm{\forall }{g}^{\prime }\in G(g^{\prime }g{g}^{\prime -1}=g)\}$, $\in \{\text{ the normal subgroups of }G\}$
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Conditions:
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2: Note

Let us see that $Z(G)$ is indeed a normal subgroup of $G$.

For each $g_1,g_2\in Z(G)$, $g_1{g}_2\in Z(G)$, because for each $g^{\prime }\in G$, $g^{\prime }{g}_1{g}_2{g}^{\prime -1}=g^{\prime }{g}_1{g}^{\prime -1}{g}^{\prime }{g}_2{g}^{\prime -1}=g_1{g}_2$. $1\in Z(G)$, because for each $g^{\prime }\in G$, $g^{\prime }1g^{\prime -1}=1$. For each $g\in Z(G)$, $g^{-1}\in Z(G)$, because as $g^{\prime }g{g}^{\prime -1}=g$, $(g^{\prime }g{g}^{\prime -1}{)}^{-1}=g^{-1}$, but $(g^{\prime }g{g}^{\prime -1}{)}^{-1}=g^{\prime }{g}^{-1}{g}^{\prime -1}$. The associativity holds because it holds in the ambient $G$. So, $Z(G)$ is a subgroup of $G$.

For each $g^{\prime }\in G$, $g^{\prime }Z(G)g^{\prime -1}=Z(G)$, because for each $g\in Z(G)$, $g^{\prime }g{g}^{\prime -1}=g$, by the definition of center of group.

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References

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