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 $\struct {\Z_4, +_4}$:

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

However, for example:
 * $\eqclass 1 4 +_4 \eqclass 1 4 = \eqclass 2 4$

and so $\eqclass 1 4$ is not self-inverse.

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

Hence the result.