## An infinite group with itself as the inverse?

Forum for the GRE subject test in mathematics.
Hom
Posts: 39
Joined: Sat Oct 01, 2011 3:22 am

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

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?

blitzer6266
Posts: 61
Joined: Sun Apr 04, 2010 1:08 pm

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

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

Hom
Posts: 39
Joined: Sat Oct 01, 2011 3:22 am

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

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.