Klein Four-Group and Group of Cyclic Group of Order 4 are not Isomorphic

Theorem
The Klein $4$-group $K_4$ and the cyclic group of order $4$ $C_4$ are not isomorphic.

Proof
Recall the Cayley table for $K_4$:

From Finite Cyclic Group is Isomorphic to Integers under Modulo Addition, $C_4$ can be exemplified using the additive group of integers modulo $4$.

Recall the Cayley table for $\left({\Z_4, +_4}\right)$:

Note that all elements of $K_4$ are self-inverse.

However, for example:
 * $\left[\!\left[{1}\right]\!\right]_4 +_4 \left[\!\left[{1}\right]\!\right]_4 = \left[\!\left[{2}\right]\!\right]_4$

and so $\left[\!\left[{1}\right]\!\right]_4$ is not self-inverse.

Thus there can be no bijection between $K_4$ and $\left({\Z_4, +_4}\right)$ such that:
 * $\forall a, b \in K_4: \phi \left({a}\right) +_4 \phi \left({b}\right) = \phi \left({a b}\right)$

Hence the result.