Homomorphism examples

Which of the following function(s) define group homomorphisms?

I. $f \colon GL_2({\mathbb R}) \to GL_2({\mathbb R})$ defined by $f(A) = A^2$

II. $f \colon {\mathbb Z}_4 \to {\mathbb Z}$ defined by $f(0) = 0,$ $f(1) = 1,$ $f(2) = 2,$ $f(3) = 3$

III. $f \colon {\mathbb Z} \to {\mathbb Z}_4$ defined by $f(x) = x \pmod 4.$

Notation:

• $GL_2({\mathbb R})$ is the group of invertible $2 \times 2$ matrices with real entries, with the operation being matrix multiplication.

• $\mathbb Z$ is the additive group of the integers.

• ${\mathbb Z}_4$ is the additive group of the integers modulo 4, whose elements are $\{0,1,2,3\}.$