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### An infinite group with itself as the inverse?

Posted: Sun Oct 16, 2011 12:04 am
Is it possible to prove or disprove the existent of such an infinite group (G,*), that for any element a belongs to G, a*a = e?

### Re: An infinite group with itself as the inverse?

Posted: Sun Oct 16, 2011 12:26 am
Consider the infinite binary sequences. In other words, an element of G is something like (0, 0, 1, 0, 1,1, ...)

The group addition would be adding component wise. In this case, the 0 sequence would be the identity, and every element has order 2.

You can also think of this as (Z/2Z)^N

where Z is the integers and N is the naturals

### Re: An infinite group with itself as the inverse?

Posted: Sun Oct 16, 2011 1:27 am
blitzer6266 wrote:Consider the infinite binary sequences. In other words, an element of G is something like (0, 0, 1, 0, 1,1, ...)

The group addition would be adding component wise. In this case, the 0 sequence would be the identity, and every element has order 2.

You can also think of this as (Z/2Z)^N

where Z is the integers and N is the naturals
wow. that's a really good example. thank you very much.