definition of center of group
Topics
About: group
The table of contents of this article
Starting Context
- The reader knows a definition of group.
Target Context
- The reader will have a definition of center 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:
\( G\): \(\in \{\text{ the groups }\}\)
\(*Z (G)\): \(= \{g \in G \vert \forall g' \in G (g' g g'^{-1} = g)\}\), \(\in \{\text{ the normal subgroups of } G\}\)
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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 \(g' \in G\), \(g' g_1 g_2 g'^{-1} = g' g_1 g'^{-1} g' g_2 g'^{-1} = g_1 g_2\). \(1 \in Z (G)\), because for each \(g' \in G\), \(g' 1 g'^{-1} = 1\). For each \(g \in Z (G)\), \(g^{-1} \in Z (G)\), because as \(g' g g'^{-1} = g\), \((g' g g'^{-1})^{-1} = g^{-1}\), but \((g' g g'^{-1})^{-1} = g' g^{-1} g'^{-1}\). The associativity holds because it holds in the ambient \(G\). So, \(Z (G)\) is a subgroup of \(G\).
For each \(g' \in G\), \(g' Z (G) g'^{-1} = Z (G)\), because for each \(g \in Z (G)\), \(g' g g'^{-1} = g\), by the definition of center of group.
