Homomorphism examples

Algebra Level 3

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

I. f ⁣:GL2(R)GL2(R) f \colon GL_2({\mathbb R}) \to GL_2({\mathbb R}) defined by f(A)=A2f(A) = A^2

II. f ⁣:Z4Z f \colon {\mathbb Z}_4 \to {\mathbb Z} defined by f(0)=0, f(0) = 0, f(1)=1, f(1) = 1, f(2)=2, f(2) = 2, f(3)=3f(3) = 3

III. f ⁣:ZZ4 f \colon {\mathbb Z} \to {\mathbb Z}_4 defined by f(x)=x(mod4). f(x) = x \pmod 4.


  • GL2(R)GL_2({\mathbb R}) is the group of invertible 2×22 \times 2 matrices with real entries, with the operation being matrix multiplication.

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

  • Z4{\mathbb Z}_4 is the additive group of the integers modulo 4, whose elements are {0,1,2,3}. \{0,1,2,3\}.


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