An infinite group with itself as the inverse?

[Hom]

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]

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]

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.