definition of normalizer of subgroup on group
Topics
About: group
The table of contents of this article
Starting Context
- The reader knows a definition of normal subgroup of group.
Target Context
- The reader will have a definition of normalizer of subgroup 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:
\( G'\): \(\in \{\text{ the groups }\}\)
\( G\): \(\in \{\text{ the subgroups of } G'\}\)
\(*N_{G'} (G)\): \(= \{g' \in G' \vert g' G g'^{-1} = G\}\)
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Conditions:
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2: Note
The name has originated from the fact that \(N_{G'} (G)\) is the largest subgroup of \(G'\) of which (\(N_{G'} (G)\)) \(G\) is a normal subgroup.
Let us confirm the fact.
Let us see that \(N_{G'} (G)\) is a group.
For each \(h_1, h_2 \in N_{G'} (G)\), \(h_1 h_2 \in N_{G'} (G)\), because \(h_1 h_2 G (h_1 h_2)^{-1} = h_1 h_2 G {h_2}^{-1} {h_1}^{-1} = h_1 G {h_1}^{-1} = G\).
\(1 \in N_{G'} (G)\), because \(1 G 1^{-1} = G\).
For each \(h \in N_{G'} (G)\), \(h^{-1} \in N_{G'} (G)\), because \(h^{-1} G h = h^{-1} (h G h^{-1}) h = (h^{-1} h) G (h^{-1} h) = 1 G 1 = G\).
Associativity holds because it holds in the ambient \(G'\).
\(G \subseteq N_{G'} (G)\), because for each \(g \in G\), \(g G g^{-1} = G\).
\(G\) is a normal subgroup of \(N_{G'} (G)\), because for each \(h \in N_{G'} (G)\), \(h G h^{-1}\).
\(N_{G'} (G)\) is such the largest, because if there is a subgroup of \(G'\), \(G''\), such that \(G\) is a normal subgroup of \(G''\), for each \(g'' \in G''\), \(g'' G g''^{-1} = G\), which means that \(g'' \in N_{G'} (G)\), which means that \(G'' \subseteq N_{G'} (G)\).
