# Math Help - quotient group

1. ## quotient group

I know that a group H has exactly one 3-Sylow subgroup L. Also, |H|=30. And |M|=15 is normal.

Now I should consider the quotient group H/M and prove that $\forall a \in H$ $a^2 \in M$. And then that each element of odd order in H is in M.

2. I don't know - it depends on what you are trying to prove...

3. Originally Posted by HoneyPi
I know that a group H has exactly one 3-Sylow subgroup L. Also, |H|=30. And |M|=15 is normal.

Now I should consider the quotient group H/M and prove that $\forall a \in H$ $a^2 \in M$. And then that each element of odd order in H is in M.

For (1): Lemma: if $N\triangleleft G\,\,\,and\,\,\,[G:N]=n$ , then $g^n\in N\,,\,\forall\,g\in G$ ( for the proof use Lagrange in the quotient $G/H$ )

For (2): Use directly Lagrange and (1).

Tonio