Normal Subgroup of order 2 (TIFR 2013 problem 10)

Question:

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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