Klein Four-Group and Group of Cyclic Group of Order 4 are not Isomorphic/Proof 2

Theorem

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

Proof

Note that both $C_4$ and $K_4$ are of order $4$.

Also note that both $C_4$ and $K_4$ are abelian.

By definition, $C_4$ has elements of order $4$.

From Order of Isomorphic Image of Group Element, the image of an element of $C_4$ under an isomorphism to $K_4$ would also be of order $4$.

But $K_4$ has no elements of order $4$.

Hence $C_4$ and $K_4$ are not isomorphic.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.3$. Isomorphism: Example $138$