# Homomorphism examples

Algebra Level 5

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 $$\{0,1,2,3\}.$$

×