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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Thu, 8 Mar 2012 21:01:53 UTC
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daveyinaz wrote:
DgrayMan wrote:
If G=<a> and b is in G, prove the order of b is a factor of the order of a.

This is what ive tried coming up with.

I know if G={a0,a1,a2...a(n-1)}
then ord(a)=n

so for some k integer b=a^k

so ord(b)=n^k which is a factor of the order of a.

I understand how this is true but again its difficult for me trying to prove for all elements in a group.

I hope you didn't really mean to say the order of because this is not a factor of the order of ...

Yeah, he got his exponent rules messed up and forgot that order is the minimal thing.

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Thu, 8 Mar 2012 21:42:52 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
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hmm. why wouldn't it be true? Since its not, how is it proven that <a>=<b>?

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Fri, 9 Mar 2012 00:25:38 UTC
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DgrayMan wrote:
hmm. why wouldn't it be true? Since its not, how is it proven that <a>=<b>?

Well for one, , and for another, I keep telling you there is no reason to believe <a>=<b> for any choice of b, I mean what if b=e?

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Fri, 9 Mar 2012 00:51:34 UTC
 S.O.S. Oldtimer

Joined: Sat, 16 Aug 2008 04:47:19 UTC
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DgrayMan wrote:
hmm. why wouldn't it be true? Since its not, how is it proven that <a>=<b>?

You could always create an isomorphism between and .

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Fri, 9 Mar 2012 01:06:46 UTC
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daveyinaz wrote:
DgrayMan wrote:
hmm. why wouldn't it be true? Since its not, how is it proven that <a>=<b>?

You could always create an isomorphism between and .

If the order of b is not the order of a, then none such exists.

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Fri, 9 Mar 2012 03:55:24 UTC
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Joined: Sat, 21 Jan 2012 03:59:22 UTC
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Just wondering, cause my teacher said if they are subsets of each other then they are equal.

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Fri, 9 Mar 2012 03:59:26 UTC
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DgrayMan wrote:
Just wondering, cause my teacher said if they are subsets of each other then they are equal.

Yes, that's true, but there's a difference between saying that and saying for an arbitrary b this is true. There are some b for which it is true, and some for which it is not.

It's of course true that if you can show then they're equal, but there was nothing in the original thing about that, you just had an arbitrary , and you were asking about the order of b, which is only marginally related to whether or not b is another generator of the group.

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