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Abstract Algebra

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Cosets and identifying normal subgroups

ABSALG-ZNU3D4

Let $G$ be a group and let $H$ be a subgroup of $G$.

$H$ is a normal subgroup of $G$ if and only if:

$gH=Hg$, for all $g\in G$

Denote:

by ${\rm GL}(2,{\mathbb R})$ the group of $2\times 2$ matrices with non-zero determinant

by ${\rm SL}(2,{\mathbb R})$ the group of $2\times 2$ matrices with determinant equal 1

Which of the following are true?

A

Let:

$H=\left\{\begin{pmatrix}1&n\cr0&1\end{pmatrix}\mid n\in{\mathbb Z}\right\}$

...and:

let $g\in {\rm GL}(2,{\mathbb R})$ be fixed

..then $gHg^{-1}\subseteq H$ always implies $gHg^{-1}=H$

B

Let $G={\rm GL}(2,{\mathbb R})$ and $H={\rm SL}(2,{\mathbb R})$ then:

...there are only finitely many distinct left cosets of $H$ in $G$.

C

Let $G={\rm GL}(2,{\mathbb R})$, $H={\rm SL}(2,{\mathbb R})$, and $g\in {\rm GL}(2,{\mathbb R})$ then:

$gHg^{-1}\subseteq H$ implies $gHg^{-1}=H$

D

Let $G$ be a finite group and $H$ a subgroup of $G$. Then for $g\in G$:

$gHg^{-1}\subseteq H$ implies $gHg^{-1}=H$

E

For every group $G$, a subgroup $H$ of $G$ is normal in $G$:

...if and only if $gHg^{-1}\subseteq H$ for all $g\in G$