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IMC 1996 Problem 9

Source: IMC 1996

March 4, 2021

group theoryabstract algebralinear algebra

Problem Statement

Let

G

be the subgroup of

GL_{2}(\mathbb{R})

generated by

A

and

B

, where

A=\begin{pmatrix} 2 &0\\ 0&1 \end{pmatrix},\; B=\begin{pmatrix} 1 &1\\ 0&1 \end{pmatrix}

. Let

H

consist of the matrices

\begin{pmatrix} a_{11} &a_{12}\\ a_{21}& a_{22} \end{pmatrix}

in

G

for which

a_{11}=a_{22}=1

. a) Show that

H

is an abelian subgroup of

G

. b) Show that

H

is not finitely generated.

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