Abstract Algebra: Theory and Applications · Problem 10.4.4

Homomorphisms — Thomas Judson

Mathematics — Abstract Algebra · normal-subgroups-factor-groups

Problem

Let

T

be the group of nonsingular upper triangular

2 \times 2

matrices with entries in

\mathbb{R}

; that is, matrices of the form

\begin{pmatrix} a & b \\ 0 & c \end{pmatrix},

where

a, b, c \in \mathbb{R}

and

ac \neq 0

. Let

U

consist of matrices of the form

\begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix},

where

x \in \mathbb{R}

. (a) Show that

U

is a subgroup of

T

. (b) Prove that

U

is abelian. (c) Prove that

U

is normal in

T

. (d) Show that

T/U

is abelian. (e) Is

T

normal in

GL_2(\mathbb{R})