## Generators of Infinite Groups

**Moderators:** gmalivuk, Moderators General, Prelates

### Generators of Infinite Groups

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

### Re: Generators of Infinite Groups

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.

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.

- Talith
- Proved the Goldbach Conjecture
**Posts:**848**Joined:**Sat Nov 29, 2008 1:28 am UTC**Location:**Manchester - UK

### Re: Generators of Infinite Groups

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.

- jestingrabbit
- Factoids are just Datas that haven't grown up yet
**Posts:**5967**Joined:**Tue Nov 28, 2006 9:50 pm UTC**Location:**Sydney

### Re: Generators of Infinite Groups

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?

ameretrifle wrote:Magic space feudalism is therefore a viable idea.

### Re: Generators of Infinite Groups

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.

- Talith
- Proved the Goldbach Conjecture
**Posts:**848**Joined:**Sat Nov 29, 2008 1:28 am UTC**Location:**Manchester - UK

### Re: Generators of Infinite Groups

Oops missed that.

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