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 Post subject: group theory questionPosted: Sat, 13 Feb 2010 20:01:20 UTC
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Joined: Mon, 2 Nov 2009 19:59:34 UTC
Posts: 96
"Let G be a nontrivial group with no proper subgroups except the trivial one. Show that G is finite and that the order of G is prime."

Okay, I think that maybe we know that G is finite because if it was infinite it you could always have an infinite subgroup that operations could not leave? I'm not sure how to prove that and I am totally stumped about how to prove that the order of G is prime.

I Know that if the order of G is even there must be an element such that where is the identity element. Therefore you could have a subgroup with just . But this only proves that finite group with no proper subgroups must have odd order, not necessary prime.

Any help would be appreciated.

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 Post subject: Re: group theory questionPosted: Sun, 14 Feb 2010 06:35:39 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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Zeta wrote:
"Let G be a nontrivial group with no proper subgroups except the trivial one. Show that G is finite and that the order of G is prime."

Okay, I think that maybe we know that G is finite because if it was infinite it you could always have an infinite subgroup that operations could not leave? I'm not sure how to prove that and I am totally stumped about how to prove that the order of G is prime.

I Know that if the order of G is even there must be an element such that where is the identity element. Therefore you could have a subgroup with just . But this only proves that finite group with no proper subgroups must have odd order, not necessary prime.

Any help would be appreciated.

Pick any non-identity element g. If it has infinite order, then is a nontrivial proper subgroup of G. If g has finite order, look at the subgroup and you get , hence must be a prime.

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 Post subject: Posted: Sun, 14 Feb 2010 06:45:27 UTC
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Also, there is a finite simple group of order 2.

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 Post subject: Re: group theory questionPosted: Sun, 14 Feb 2010 17:20:27 UTC
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Joined: Mon, 2 Nov 2009 19:59:34 UTC
Posts: 96
outermeasure wrote:

Pick any non-identity element g. If it has infinite order, then is a nontrivial proper subgroup of G. If g has finite order, look at the subgroup and you get , hence must be a prime.

Thank you for the speedy reply but I don't yet understand it. Using the set of integers under addition as an example of an infinite group, lets say that g=4. Than:

So than if we do 8*8 we get 16 which leaves the subgroup, also, This group does not contain the inverse of any of these numbers.

I'm sure I'm probably missing a fundamental fact about what you mean that will clear this up but please help.

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 Post subject: Posted: Tue, 16 Feb 2010 20:00:57 UTC
 S.O.S. Oldtimer

Joined: Sat, 16 Aug 2008 04:47:19 UTC
Posts: 208
Consider the group of integers under addition, when g = 4, then

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 Post subject: Posted: Tue, 16 Feb 2010 20:40:19 UTC
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Joined: Mon, 2 Nov 2009 19:59:34 UTC
Posts: 96
daveyinaz wrote:
Consider the group of integers under addition, when g = 4, then

Thank you! That clears things up.

I had thought that might be it but it's nice to know for sure.

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