723: Reversed Operator Group of Group

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definition of reversed operator group 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: 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 reversed operator group 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:
$G$: $\in \{\text{ the groups }\}$, with the operator, $\cdot$
$\ast \tilde{G}$: $\in \{\text{ the groups }\}$, $=G$ as the set, with the operator, $\ast$
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Conditions: $\mathrm{\forall }{p}_1,p_2\in \tilde{G}(p_1\ast p_2=p_2\cdot p_1)$
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2: Natural Language Description

For any group, $G$, with the operator, $\cdot$, the group, $\tilde{G}$, with $G$ as the set and $\ast$ as the operator such that for each $\mathrm{\forall }{p}_1,p_2\in \tilde{G}$, $p_1\ast p_2=p_2\cdot p_1$

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3: Note

$\tilde{G}$ is indeed a group: 0) for each $p_1,p_2\in \tilde{G}$, $p_1\ast p_2=p_2\cdot p_1\in G=\tilde{G}$; 1) for each elements, $p_1,p_2,p_3\in \tilde{G}$, $(p_1\ast p_2)\ast p_3=p_3\cdot (p_2\cdot p_1)=(p_3\cdot p_2)\cdot p_1=p_1\ast (p_2\ast p_3)$; 2) the identity element, $1\in G$, is the identity element in $\tilde{G}$, because for each $p\in \tilde{G}$, $1\ast p=p\cdot 1=p=1\cdot p=p\ast 1$, so, the notation, "$1$", is used without specifying which group it is the identity element of; 3) for each $p\in \tilde{G}$, the inverse, $p^{-1}\in G$, is the inverse in $\tilde{G}$, because $p^{-1}\ast p=p\cdot p^{-1}=1=p^{-1}\cdot p=p\ast p^{-1}$, so, the notation, "$p^{-1}$", is used without specifying which group it is the inverse in.

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References

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