description/proof of that subgroup of Abelian group is normal subgroup
Topics
About: group
The table of contents of this article
Starting Context
- The reader knows a definition of Abelian group.
- The reader knows a definition of normal subgroup of group.
Target Context
- The reader will have a description and a proof of the proposition that any subgroup of any Abelian group is a normal subgroup.
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:
\(G'\): \(\in \{\text{ the Abelian groups }\}\)
\(G\): \(\in \{\text{ the subgroups of } G'\}\)
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Statements:
\(G \in \{\text{ the normal subgroups of } G'\}\)
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2: Proof
Whole Strategy: Step 1: take any \(g' \in G'\), and see that \(g' G g'^{-1} = G\).
Step 1:
Let \(g' \in G'\) be any.
Let us see that \(g' G g'^{-1} = G\).
Let \(g' g g'^{-1} \in g' G g'^{-1}\) be any.
\(g' g g'^{-1} = g' g'^{-1} g = g \in G\), so, \(g' G g'^{-1} \subseteq G\).
Let \(g \in G\) be any.
\(g = g' g'^{-1} g = g' g g'^{-1} \in g' G g'^{-1}\), so, \(G \subseteq g' G g'^{-1}\).
So, \(g' G g'^{-1} = G\).
