Thread: More questions in Abstract-Help is needed!

1. 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 $[S_{n}$ 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!

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

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

3. Thanks a lot