Abstract Algebra: Theory and Applications · Problem 11.4.2

Matrix Groups and Symmetry — Thomas Judson

Mathematics — Abstract Algebra · identifying-homomorphisms-kernel

Problem

Which of the following maps are homomorphisms? If the map is a homomorphism, what is the kernel? (a)

\phi : \mathbb{R}^* \to GL_2(\mathbb{R})

defined by

\phi(a) = \begin{pmatrix} 1 & 0 \\ 0 & a \end{pmatrix}

(b)

\phi : \mathbb{R} \to GL_2(\mathbb{R})

defined by

\phi(a) = \begin{pmatrix} 1 & 0 \\ a & 1 \end{pmatrix}

(c)

\phi : GL_2(\mathbb{R}) \to \mathbb{R}

defined by

\phi\left(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\right) = a + d

(d)

\phi : GL_2(\mathbb{R}) \to \mathbb{R}^*

defined by

\phi\left(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\right) = ad - bc

(e)

\phi : \mathbb{M}_2(\mathbb{R}) \to \mathbb{R}

defined by

\phi\left(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\right) = b,

where

\mathbb{M}_2(\mathbb{R})

is the additive group of

2 \times 2

matrices with entries in

\mathbb{R}

. Hint. (a) is a homomorphism with kernel

\{1\}

; (c) is not a homomorphism.