Finite group with conjugacy class of order 2 has nontrivial normal subgroup?

Question

Let $G$ be a finite group with a conjugacy class of order 2. How do I go about showing that $G$ has a nontrivial normal (proper) subgroup?

Let $a$ be in the conjugacy class of order 2. Then $2 = [G:N(a)]$, so $G$ has even order. Can I use that somehow?

You are done here: since the two cosets of $N(a)$ form a group, $N(a)$ is a normal subgroup of $G$. — Karl Kroningfeld, Apr 15 '13 at 10:54
In fact, if $[G:N(a)]=2$ for some $a\in G$, then the right coset of $N(a)$ in the group has 2 distinct elements. — Mikasa, Apr 15 '13 at 10:56

score 2 · Accepted Answer · answered Apr 15 '13 at 11:08

2

Hint:

Can you prove that a subgroup of index $\,2\,$ in any group is always normal ?

answered Apr 15 '13 at 11:08

DonAntonio

1 Answers1