In particular, when n = 2, is an abelian group since for any group element , is of order or , if is of order , then is isomorphic to cyclic group of order , hence abelian. On the other hand, if any non-trivial element in is of order , hence by the conclusion above, then or . We only need to consider the case when , then there is an element of which is not in the center of . Note that is of order , so the subgroup of generated by contains elements and thus is a proper subset of , because includes all elements of this subgroup and the center which does not contain but at least elements. Hence the order of is strictly larger than , therefore = , therefore is an element of the center of . Hence is abelian and in fact isomorphic to the direct product of two cyclic groups each of order .