Prove $\mathbb{Z}_4\ncong \mathbb{Z}_2\times\mathbb{Z}_2$
Solution 1:
Let $f:\mathbb Z_4 \to \mathbb Z_2 \times\mathbb Z_2$ be a group morphism, with $f(1)=x$. Then $f(2)=f(1)+f(1)=x+x=0=f(0)$, whence $f$ cannot be an isomorphism.
Prove $\mathbb{Z}_4\ncong \mathbb{Z}_2\times\mathbb{Z}_2$
Solution 1:
Let $f:\mathbb Z_4 \to \mathbb Z_2 \times\mathbb Z_2$ be a group morphism, with $f(1)=x$. Then $f(2)=f(1)+f(1)=x+x=0=f(0)$, whence $f$ cannot be an isomorphism.