definition of reversed operator group of group
Topics
About: group
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of group.
Target Context
- The reader will have a definition of reversed operator group of group.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( G\): \(\in \{\text{ the groups }\}\), with the operator, \(\cdot\)
\(*\widetilde{G}\): \(\in \{\text{ the groups }\}\), \(= G\) as the set, with the operator, \(*\)
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Conditions: \(\forall p_1, p_2 \in \widetilde{G} (p_1 * p_2 = p_2 \cdot p_1)\)
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2: Natural Language Description
For any group, \(G\), with the operator, \(\cdot\), the group, \(\widetilde{G}\), with \(G\) as the set and \(*\) as the operator such that for each \(\forall p_1, p_2 \in \widetilde{G}\), \(p_1 * p_2 = p_2 \cdot p_1\)
3: Note
\(\widetilde{G}\) is indeed a group: 0) for each \(p_1, p_2 \in \widetilde{G}\), \(p_1 * p_2 = p_2 \cdot p_1 \in G = \widetilde{G}\); 1) for each elements, \(p_1, p_2, p_3 \in \widetilde{G}\), \((p_1 * p_2) * p_3 = p_3 \cdot (p_2 \cdot p_1) = (p_3 \cdot p_2) \cdot p_1 = p_1 * (p_2 * p_3)\); 2) the identity element, \(1 \in G\), is the identity element in \(\widetilde{G}\), because for each \(p \in \widetilde{G}\), \(1 * p = p \cdot 1 = p = 1 \cdot p = p * 1\), so, the notation, "\(1\)", is used without specifying which group it is the identity element of; 3) for each \(p \in \widetilde{G}\), the inverse, \(p^{-1} \in G\), is the inverse in \(\widetilde{G}\), because \(p^{-1} * p = p \cdot p^{-1} = 1 = p^{-1} \cdot p = p * p^{-1}\), so, the notation, "\(p^{-1}\)", is used without specifying which group it is the inverse in.
