What are the orders of (x,v) and (x,u²) in A x B?

feyomi

If A = {(x,y) : x³ = y² = e, yx = xˉ¹y} and B = {(u,v) : u^4 = v² = e, vu = uˉ¹v}, how do I go about finding the orders of, say, (x,v) and (x,u²) in A x B?

Thanks.

DonAntonio

Your writing seems to be wrong: as you put it, both A and B are sets of ordered pairs, but then again you write elements of A x B as pairs of elements, NOT of pairs...I think you meant to write that A, B are groups, A generated by x, y and B by u,v with the given relations.
Then in A x B, both elements (x,v), (x, u^2) clearly have order dividing 6. The only thing left to do is to actually research deeper what exactly x,u,v,y are within those groups...(for example, in A the element of order dividing 3 is conjugate to its inverse, whereas in B the element of order dividing 4 is conjugate to its inverse...what does this mean?)