Suppose that G is a group with where is a prime and . Prove that is not simple.

- Feb 28th 2011, 03:30 PMJJMC89G is a group with |G|=mp where p is a prime and 1<m<p. Prove that G is not simple.
Suppose that G is a group with where is a prime and . Prove that is not simple.

- Feb 28th 2011, 06:17 PMTinyboss
How many Sylow p-subgroups can G have?

- Feb 28th 2011, 06:44 PMJJMC89
- Feb 28th 2011, 06:54 PMTinyboss
There are always Sylow p-subgroups. In this case, it follows immediately from Sylow's theorems that there is a unique one. Unique Sylow subgroups are normal.

But as you say, you haven't covered that yet, so I'm sure you're expected to make a more direct argument. - Feb 28th 2011, 08:01 PMJJMC89
Since , there is a a subgroup of order . A (sub)group of order is cyclic, thus is Abelian thus normal. Since we have a normal subgroup and , is not simple.

Is this argument correct? - Feb 28th 2011, 08:04 PMTinyboss
Abelian subgroups need not be normal. But I think you're on the right track considering the cyclic subgroup generated by an order-p element.

- Mar 1st 2011, 12:32 PMJJMC89
- Mar 1st 2011, 11:26 PMDrexel28

Let be such that . It is trivial that there is a homomorphism by having . Moreover, one can prove that . Now, since is prime and we must have that or . Suppose that then is a subgroup of of order and so but since is prime and this is impossible. Thus, and so so that isn't simple.