.I have a huge test tommorow on this subject, and I'm pretty bad at it...Hope you'll be able to help me...
1. Prove that doesn't have any subgroup of index t where 2<t<n and n>=5.
Suppose is a sbgp. of index . Then acts on the set of left cosets of (the regular representation), and this determines a homomorphism whose kernel is the core of the normal subgroup of which is maximal wrt to being contained in . Since we know has no normal sbgps. but the trivial ones and , and is none of this (why?), we get that the above kernel must be trivial is embedded in , which of course is absurd (why?)
2. Let G be an abelian finite group which isn't cyclic. Prove that there is a prime number p such as G contains a subgroup that is isomorphic to CpxCp.
You can first prove this for a p-group G and then generalize, say by induction, to any finite order
3. Prove that every group G of order 30 contains a subgroup of order 15.
How many Sylow 3-sbgps. and 5-sbgps. does such a group G can have?
Thanks a lot!