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 Post subject: elementary Properties of cyclic subgroupsPosted: Thu, 8 Mar 2012 05:35:02 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
Posts: 182
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.

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Thu, 8 Mar 2012 05:41:48 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
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.

Lagrange's theorem.

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Thu, 8 Mar 2012 05:58:39 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
Posts: 182
Great, the proof uses cosets and we haven't even discussed cosets! This assigned textbook is very backwards. But how does my interpretation hold since i don't really know what cosets are?

Last edited by DgrayMan on Thu, 8 Mar 2012 06:01:05 UTC, edited 1 time in total.

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Thu, 8 Mar 2012 06:00:38 UTC
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DgrayMan wrote:
Great, the proof uses cosets and we haven't even discussed cosets! This assigned textbook is very backwards.

Have you not learned Lagrange yet? If so, then use the fact that for some k. Then you know that, if n is the order of a, then , so the order of b divides n.

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Thu, 8 Mar 2012 06:03:02 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
Posts: 182
No i havent heard it used in this course yet. The book we are using is " A book of absract algebra" 2nd edition by charles c. pinter. The chapter of cosets is two chapters away from where we are at.

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Thu, 8 Mar 2012 06:05:17 UTC
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DgrayMan wrote:
No i havent heard it used in this course yet. The book we are using is " A book of absract algebra" 2nd edition by charles c. pinter. The chapter of cosets is two chapters away from where we are at.

Ah, then just use the other argument I said.

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Thu, 8 Mar 2012 06:06:21 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
Posts: 182
Do you recommend any books on abstract algebra that would be helpful?

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Thu, 8 Mar 2012 06:09:10 UTC
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DgrayMan wrote:
Do you recommend any books on abstract algebra that would be helpful?

The book I learned out of was Herstein, but I can hardly recommend that one to you. I have seen good things from Rotman as well as Dummit and Foote though, maybe check those out, though from what I understand they're primarily geared at graduate level things, so they might not be as good either. My undergraduate location used Nicholson for the undergrads, so that's what I'd say you should check out. I haven't used it myself, but the university was always good to me, so I'll put my faith in it.

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Thu, 8 Mar 2012 06:10:20 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
Posts: 182
also, does this prove <a> is a subset of <b>?

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Thu, 8 Mar 2012 06:12:14 UTC
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DgrayMan wrote:
also, does this prove <a> is a subset of <b>?

Hmm? No, that's not true <b> is a subset of <a>, since <b> is a subset of G which is equal to <a>.

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Thu, 8 Mar 2012 06:14:55 UTC
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Joined: Sat, 21 Jan 2012 03:59:22 UTC
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ah i see, it would have to be a=b^k for <a> to be a subset of <b>

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Thu, 8 Mar 2012 06:29:44 UTC
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DgrayMan wrote:
ah i see, it would have to be a=b^k for <a> to be a subset of <b>

yes.

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Thu, 8 Mar 2012 06:54:33 UTC
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Joined: Sat, 21 Jan 2012 03:59:22 UTC
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Now by proving that <a> is a subset of <b> and <b> is a subset of <a>, does this mean <a>=<b>?

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Thu, 8 Mar 2012 15:54:53 UTC
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DgrayMan wrote:
Now by proving that <a> is a subset of <b> and <b> is a subset of <a>, does this mean <a>=<b>?

If you can manage it, yes, but you realize that it probably isn't true, right?

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 Post subject: Re: elementary Properties of cyclic subgroupsPosted: Thu, 8 Mar 2012 18:10:47 UTC
 S.O.S. Oldtimer

Joined: Sat, 16 Aug 2008 04:47:19 UTC
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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 ...

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