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**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 $\displaystyle [S_{n}$ doesn't have any subgroup of index t where 2<t<n and n>=5.

Suppose $\displaystyle H\leq S_n\,,\,\,n\geq 5$ is a sbgp. of index $\displaystyle t\,,\,\,2<t<n$. Then $\displaystyle S_n$ acts on the set of left cosets of $\displaystyle S_n\,\,\,in\,\,\,H$ (the regular representation), and this determines a homomorphism $\displaystyle S_n\rightarrow S_t$ whose kernel is the core of $\displaystyle H=$ the normal subgroup of $\displaystyle S_n$ which is maximal wrt to being contained in $\displaystyle H$ . Since we know $\displaystyle S_n$ has no normal sbgps. but the trivial ones and $\displaystyle A_n$ , and $\displaystyle H$ is none of this (why?), we get that the above kernel must be trivial $\displaystyle \Longrightarrow\,\, S_n$ is embedded in $\displaystyle S_t$, 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!