Generators of Infinite Groups

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Timefly
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Generators of Infinite Groups

Postby Timefly » Wed Sep 21, 2011 4:45 pm UTC

Is there any generalized method of finding out if an infinite group has a generator.

letterX
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Re: Generators of Infinite Groups

Postby letterX » Wed Sep 21, 2011 5:13 pm UTC

What exactly are you asking: are you asking to find some single generator that's part of a larger generating set? In that case, every group has a generating set (just take the whole set) which contains any specified element, so this is sort of uninterestingly true.

Or perhaps you're asking if there is a single generator which generates the whole group. In which case your group is infinite cyclic, so is isomorphic to Z. Then, I'd say that "telling if some arbitrary group you've just been handed is actually Z" depends on what format it's given to you in.

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Talith
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Re: Generators of Infinite Groups

Postby Talith » Wed Sep 21, 2011 5:31 pm UTC

Asking whether a group is generated by a finite subset of elements is a much more interesting problem and one which has been completely solved for the Abelian case: see Finitely generated Abelian groups.
Last edited by Talith on Wed Sep 21, 2011 6:26 pm UTC, edited 1 time in total.

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jestingrabbit
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Re: Generators of Infinite Groups

Postby jestingrabbit » Wed Sep 21, 2011 5:43 pm UTC

Talith wrote:Asking whether a group is generated by a finite subset of elements is a much more interesting problem and one which has been completely solved:


In the Abelian case or the general case?
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letterX
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Re: Generators of Infinite Groups

Postby letterX » Wed Sep 21, 2011 5:47 pm UTC

jestingrabbit wrote:
Talith wrote:Asking whether a group is generated by a finite subset of elements is a much more interesting problem and one which has been completely solved:


In the Abelian case or the general case?

Indeed. In the general case, I'd be much more surprised if there were an answer. You tend to run into nasty problems like "If I hand you a group given by a finite number of generators and relations, decide if it's the trivial group or not". This turns out to be undecidable.

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Talith
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Re: Generators of Infinite Groups

Postby Talith » Wed Sep 21, 2011 6:26 pm UTC

Oops missed that.


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