Show that every group of order 4 must be isomorphic to Z4 or Z2 x Z2?

Suppose G= {a, b, c, e} is a group of order 4 where e is the identity element.
Since G has order 4, a,b,c,e are distinct.

If G is cyclic, then G is isomorphic to Z4.

If G is not cyclic, consider an element say, a, which is not the identity element.
the order of the element a must divide 4(order of the group). hence its order is either 1,2 or 4

    if the order is 4 then 'a' is a generator of G thus G is cyclic(contradiction)

    if the order is 1 then 'a' is the identity element. (contradiction).

    Hence, the order of 'a' must be 2

    Every non-identity element of G has order 2, hence G is isomorphic to Z2 x Z2