Assume $G$ is generated by a finite number of elements which are all self-inverse in $G$. Is it possible to make any assumptions about the order of $G$?
Take for example, $G = \langle a,b\rangle :a^2 = b^2 = e$, is it true that we cannot make any assumption on the order of the element $abab…$ and thus cannot assume the order of G? Am I missing something? Is there a non-trivial condition we might ask for to help us?
Note: to clarify, only the generating elements are self-inverse in G.
Best Answer
If a group is generated by a finite number of elements of order $2$ all we can determine is that it has at most countable order, and that its order must be even when finite.
Clearly if the order is finite it must be even, because of Lagrange's theorem.
Notice that the dihedral group $D_{2n}$ is generated by its elements of order $2$, and has order $2n$, so every even order is possible.
Finally notice that since the number of generators is finite, the group must have at most countable order.