# 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.