Translation: If $G$ is a finite group in which every element $g \in G$ satisfies $g^2 = e$, where $e$ is the unit element of $G$, then what are the possible values for the order $k=|G|$ of $G$?
Original: There is a group with $k$ elements $G = (g_{i})$, for all elements of which $g_{i}^{2} = e$, where $e$ is the unit element. How to get the order of the group (the properties of the elements of the group leads to restrictions on the number of k)?
You might note that $a^2=e$ and $b^2=e$ and $(ab)^2=abab=e$. From this last, using the associative law we get $a(abab)b=a(e)b=ab$
Then $a^2bab^2=ebae=ba=ab$
So the group is abelian of exponent 2. Do you know any structure theorem about such groups which might suggest the possible orders?