Hi, it's my first time posting in this forum, so I'm sorry if I have done anything against the forum rules and please point it out to me. Currently revising group theory for an exam in a week's time, and these two practise questions I couldn't finish, so if anyone can push me towards the right direction it'll be great.(adsbygoogle = window.adsbygoogle || []).push({});

The problem statement, all variables and given/known data

1. Prove that a simple abelian group G is cyclic of prime order.

2. Let G be a simple group and have a subgroup of index k > 2. Prove that |G| divides k!/2.

The attempt at a solution

Question 1

Suppose G is simple abelian, and consider an element x in G where x is not the identity element. Then since G is abelian, the cyclic group generated by x, <x> is normal in G, and this implies that G = <x>, so G is cyclic.

But now I'm not sure how to prove the fact that G is finite and has prime order...

Question 2

Suppose H is a subgroup of G, then by LaGrange |G| = |H||G| = |H|k. Also since k > 2, clearly k!/2 = k*(k-1)*(k-2)*...*3. So |G| divides k!/2 for any k > 2.

But I didn't even use the fact that G is simple, and this argument seems far too simple and I don't feel confident with the solution...

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# Homework Help: Questions involving simple groups

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