Infinite groups with all elements of order 2?

Question

If G is a group such that $a^2 =e$ for all $a \in G$, where $e$ is the identity element in $G$, then $G$ is finite.

This question can be proved false if we can get a group of infinite order with each non-identity element of order 2.

I thought of Klien's-4 group (order of each element=2) & Group of n-th root of unity (order of each element=n). But neither example works here.

Please help me in finding the counterexample for the above statement.

Answer 1

Extend the idea of the Klein-$4$ group:

$$\bigoplus^{\infty} \mathbb{Z}_2$$

Answer 2

Let $S$ be a set. Then the power set $\mathcal P(S)$ is a group (with symmetric set difference as operation).

Answer 3

For example:

The product of infinitely many copies of the cyclic group of order $2$ $$C_{2}^{\omega}=C_2\times C_2\times\dots $$

Answer 4

More elementarily, the (additive) group of polynomials with coefficients in $\mathbf{F}_2$. Or equivalently in computing terms, the group of finite binary strings with the bitwise XOR operation, where in order to "add" two strings of different lengths, the shorter one is zero-padded.