What is the group $G=\langle a,b\mid a^2=b^2=1\rangle$?

Question

I am trying to describe all the elements of $G=\langle a,b\mid a^2=b^2=1\rangle$, and I am really struggling with where to start.

I feel like the group could be infinite, with each element in the group, aside from the identity, having order $2$ (I have since learnt that these are called Boolean, or Abelian-$2$, Groups - again, not sure if this is the right track to go down). I would like to go on to build two subgroups, $S_1$ and $S_2$ that are isomorphic to $\mathbb{Z} /2\mathbb{Z}$ and $\mathbb{Z}$, but obviously to do that I need to understand what $G$ actually looks like!

Thank you for any help.

Answer 1

So, how $G$ looks like. Elements of $G$ are words with letters $a,b,a^{-1},b^{-1}$ where you have to substitute $a^2$ and $b^2$ with $1$. So some elements are

$$a,b,ab,aba,abab,ababa,ababab,...$$

Note that I don't use $a^{-1}$ and $b^{-1}$ since $a=a^{-1}$ and $b=b^{-1}$

It is clear that $ab$ has infinite order.

Answer 2

Hint: $ab$ will not have order $2$ in $G$. The standard construction of the free group on $a$ and $b$ should suggest to you a normal form for elements of $G$ as words in $a$, $b$, $a^{-1}$ and $b^{-1}$ that admit no cancellation subject to the rules that $a^2 = 1 $ and $b^2 = 1$.