The starting point of this question is the following:

If $G$ is a group such that all elements have order at most $2$, then $G$ is commutative.

If $G$ is any group, let $G_{>2}$ denote the set of elements $g\in G$ such that $g^2 \neq 1_G$ where $1_G$ denotes the neutral element of the group.

Question. If $G$ is infinite and $|G_{>2}|<|G|$, is $G$ necessarily commutative?

The starting point of this question is the following:

If $G$ is a group such that all elements have order at most $2$, then $G$ is commutative.

If $G$ is any group, let $G_{>2}$ denote the set of elements $g\in G$ such that $g^2 \neq 1_G$ where $1_G$ denotes the neutral element of the group.

Question. If $G$ is infinite, directly indecomposable, and $|G_{>2}|<|G|$, is $G$ necessarily commutative?