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Problem:True/False?


Any normal subgroup of order 2 is contained in the center of the group.


Discussion:


If \(N\) is a normal subgroup of a group \(G\) and \(|N|=2\), then \(N=\left\{e,a\right \}\) where \(a^2=e\).

For all \(g\in G\) we have \(gag^{-1}\in N\).

Can \(gag^{-1}=e\)? No. Since that would imply \(a=e\).

Therefore, for all \(g\in G\), \(gag^{-1}=a\).

Which proves that a is in the center of the group.


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  • What is this topic:Abstract Algebra
  • What are some of the associated concept: Normal Subgroup, Center of a Group
  • Book Suggestions: Contemporary Abstract Algebra by Joseph A. Gallian