Homomorphism examples

Algebra Level 4

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 \( \{0,1,2,3\}.\)

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