example of a subgroup of index $3$ which is not normal

Question

Could you please give an example of a subgroup of index $3$ which is not normal ?

I know every subgroup of index $2$ is normal but if index is $3$ , I have no idea whether all of the subgroups are normal or not.

Consider subgroups of $S_3$. – Ian Coley Apr 03 '14 at 18:26 — Ian Coley, Apr 03 '14 at 18:26

score 6 · Answer 1 · answered Apr 03 '14 at 18:29

6

Hint. Consider $S_3$ and the subgroup $\langle(1\ 2)\rangle$.

answered Apr 03 '14 at 18:29

Hubble

thank you but can you be more specefic – Saeid Ghafouri Apr 03 '14 at 18:59
$H = \langle (1 \ 2) \rangle = {1, (1 \ 2)}$, that is, the subgroup generated by the transposition $(1 \ 2)$. Since the symmetric group $S_3$ contains $3! = 6$ elements and $|H| = 2$, you have that the index of $H$ is 3. Do you know how to show that a subgroup is normal? – Hubble Apr 03 '14 at 20:09

score 2 · Answer 2 · edited Apr 13 '17 at 12:21

2

As Ian Coley point out $S_3$ is smallest counter example of it.

But it is true if $G$ has odd order.

Fact: Let $p$ be smallest prime dividing $|G|$ then any group of index $p$ is normal.

Notice that when $|G|$ is odd and $3$ divides $|G|$, then $3$ is the smallest prime dividing $G$.

For fact you can check this.

edited Apr 13 '17 at 12:21

Community

answered Apr 03 '14 at 18:34

mesel

2 Answers2