definition of conjugate subgroup of subgroup by element
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 conjugate subgroup of subgroup by element.
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:
\( G'\): \(\in \{\text{ the groups }\}\)
\( G\): \(\in \{\text{ the subgroups of } G'\}\)
\( p\): \(\in G'\)
\(*p G p^{-1}\): \(= \text{ the conjugate subgroup of } G \text{ by } p\)
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Conditions:
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2: Natural Language Description
For any group, \(G'\), any subgroup, \(G\), of \(G'\), and any \(p \in G'\), \(p G p^{-1}\) is the conjugate subgroup of \(G\) by \(p\)
3: Note
\(p G p^{-1}\) is indeed a subgroup: \(1 = p 1 p^{-1} \in p G p^{-1}\); for each \(p p_1 p^{-1}, p p_2 p^{-1} \in p G p^{-1}\), \(p p_1 p^{-1} p p_2 p^{-1} = p p_1 p_2 p^{-1} \in p G p^{-1}\); for each \(p p_1 p^{-1} \in p G p^{-1}\), \(p {p_1}^{-1} p^{-1} \in p G p^{-1}\) 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.
