Clarification on the order of a group

Question

I just need some clarification. I have a question stating that G is a group and x and y are elements of that group, both of which are order 2. So being that the order of G is finite, what would be the order of xy? It can be assumed that x and y are not inverses of each other.

I think x^2 is the identity and y^2 is also the identity being as both of them are of order two.

Am I thinking about this correctly?

What you quote as what the question states, $G$ need not be finite?! Anyway, $xy$ can have arbitrary order (except infinite order if $G$ has to be finite) — Hagen von Eitzen, Aug 31 '20 at 16:24
If in addition you are given the order of $xy$ then we can say stuff about the order of your group (it divides a certain integer dependent on this number). However, just knowing that the orders of $x$ and $y$ are two is not enough. — user1729, Aug 31 '20 at 16:52

score 3 · Answer 1 · answered Aug 31 '20 at 16:38

The dihedral group $D_{2n}$ admits this presentation: $$ \langle r,s \mid r^n=s^2=(sr)^{2}=1 \rangle $$ Then $x=s$ and $y=sr$ have order $2$ but $xy=r$ has order $n$.

Therefore, the product of two elements of order $2$ can have arbitrarily high order.

Clarification on the order of a group

1 Answers1