definition of centralizer of element on 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 centralizer of element on 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:
\( G\): \(\in \{\text{ the groups }\}\)
\( p\): \(\in G\)
\(*C_G (p)\): \(= \{p' \in G \vert p' p p'^{-1} = p\}\), \(\in \{\text{ the subgroups of } G\}\)
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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 \(G\), \(C_G (p) := \{p' \in G \vert p' p p'^{-1} = p\}\)
3: Note
\(C_G (p)\) is indeed a subgroup: 0) for each \(p', p'' \in C_G (p)\), \(p' p'' p (p' p'')^{-1} = p' p'' p {p''}^{-1} p'^{-1} = p' p p'^{-1} = p\); 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 \(1 p 1^{-1} = p\); 3) for each element, the inverse is in \(C_G (p)\), because \(p'^{-1} p (p'^{-1})^{-1} = p'^{-1} p p' = p'^{-1} p' p p'^{-1} p' = p\).
