Thread: More questions in Abstract-Help is needed!

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.

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.

3. Prove that every group G of order 30 contains a subgroup of order 15.


Thanks a lot!

Originally Posted by WannaBe

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?

Tonio

Thanks a lot!

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Thanks a lot