Is every subgroup of infinite Boolean group finite

Question

Definition (Boolean group): A group $(G,*)$ is said to be Boolean if every non identity element has order $2$ $i.e$ for any $a\in G,o(a)=2$, where $a\neq e$

Now I want to know that given any infinite Boolean group $(G,*)$, is it true that every proper subgroup of $G$ is finite?

I searched on internet and found that Prüffer $p$ groups are one in which every proper subgroup is finite, but are they only group with this property being infinite, but all proper subgroups are finite)?

Answer 1

Is every subgroup of infinite Boolean group finite?

No.

Consider

$$\prod_{i=1}^\infty\Bbb Z_2.$$

The subgroup $$\prod_{i=2\\ i\text{ even}}^\infty \Bbb Z_2$$ is infinite.