Abelian Simple Group / Prime Numbers

Rederick
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Homework Statement



If G is an abelian simple group then G is isomorphic to Zp for some prime p (do not assume G is a finite group).

Homework Equations



In class, we were told an example of a simple group is a cyclic group of prime order.

The Attempt at a Solution



Let G be an abelian simple group. Let a be an element in G that is not the identity. Let H be a cyclic subgroup of G, H=<a>. Since a is an element in G, it's non-trivial. Since G is abelian, H is normal in G. Since G is a simple group, it has only 2 normal subgroups, the identity and itself. Thus, H=G. Since G is cyclic and an abelian simple group, it has prime order. Hence G is isomorphic to Zp.

I feel uneasy about the last 2 sentences. Any advice to make this a more solid proof?
 
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You have shown that whenever 1 \neq a \in G, then \langle a\rangle = G. You need one more sentence to explain why this fact shows that |G| is prime. (This sentence should replace "Since G is cyclic and an abelian simple group, it has prime order", which -- though you've added the observation that G is cyclic -- is what you're trying to prove.)
 
Yes, the last 2 sentences are a bit to fast. So far you've only shown that a simple abelian group must be cyclic. You must still show now that cyclic simple groups are of prime order.

How do we show that? Take a cyclic group that is not of prime order. Try to find a proper subgroup in there. This shows that our group is not simple...
 
an old thread, but I stumbled upon it looking for homework help so I figured I'd contribute anyway.

You must find a normal subgroup to show its not simple.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

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