There are two elements x, y in a group (G, *) such that every element in the group can be written as a product of some n...

Question

There are two elements x, y in a group (G, *) such that every element in the group can be written as a product of some number of x's and y's in some order. It is known that
  x * x = y * y = x * y * x * y = y * x * y * x = e
where e is the identity element. The maximum number of elements in such a group is _____.

Accepted Answer

The maximum number of elements is 4. The relations say x and y are each their own inverse (x*x = e, y*y = e) and that x and y commute, because (xy)(xy) = e together with x*x = y*y = e forces xy = yx. With two commuting elements of order 2, every product collapses to one of just four distinct elements: e, x, y, xy. That is the Klein four-group.