Abstract Algebra: Theory and Applications · Problem 11.4.4

Matrix Groups and Symmetry — Thomas Judson

Mathematics — Abstract Algebra · homomorphism-kernel-image

Problem

Let

\phi : \mathbb{Z} \to \mathbb{Z}

be given by

\phi(n) = 7n

. Prove that

\phi

is a group homomorphism. Find the kernel and the image of

\phi

. Hint. Since

\phi(m+n) = 7(m+n) = 7m + 7n = \phi(m) + \phi(n)

\phi

is a homomorphism.