Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 44

Exercises

Exercise

Prove the following subgroup criterion. A nonempty subset ${}H\subseteq G$ of a group ${}G$ is a subgroup if and only if the following holds:

${\text{ for all }}g,h\in H{\text{ is }}gh^{-1}\in H.$

Exercise

Let ${}G$ be a group, let ${}g\in G$ denote an element, and let ${}m,n\in \mathbb {Z}$ be integers. Show the following rules for the exponent.

1. We have ${}g^{0}=e_{G}$.
2. We have ${}g^{m+n}=g^{m}g^{n}$.

Exercise *

Show that the subgroups of ${}\mathbb {Z}$ are precisely the subsets of the form

${}\mathbb {Z} d={\left\{kd\mid k\in \mathbb {Z} \right\}}\,,$

with a uniquely determined nonnegative number ${}d$.

Exercise

Compute the order of the matrix

${\begin{pmatrix}2&4&1\\3&2&0\\0&1&3\end{pmatrix}}$

over the field ${}{\mathbb {F} }_{5}$.

Exercise

We consider the rational numbers ${}(\mathbb {Q} ,+,0)$ as commutative group. Let ${}G\subseteq \mathbb {Q}$ be a finitely generated subgroup. Show that ${}G$ is cyclic.

Exercise

Prove Lemma 44.7 .

Exercise

Let ${}G$ denote a (multiplicatively written) commutative group, and let ${}n\in \mathbb {N}$. Show that taking powers

$G\longrightarrow G,x\longmapsto x^{n},$

is a group homomorphism.

Exercise

Let ${}G$ be a commutative group, and let

$\varphi \colon G\longrightarrow H$

denote a surjective group homomorphism. Show that ${}H$ is also commutative.

Exercise

Let ${}G$ be an additively written commutative group. Show that the negation, that is, the mapping

$G\longrightarrow G,x\longmapsto -x,$

is a group isomorphism.

Exercise

Determine whether the mapping given by the floor

$\mathbb {Q} \longrightarrow \mathbb {Z} ,q\longmapsto \lfloor q\rfloor ,$

is a group homomorphism, or not.

Exercise

a) For what real polynomials ${}P\in \mathbb {R} [X]$ is the corresponding polynomial mapping

$(\mathbb {R} ,0,+)\longrightarrow (\mathbb {R} ,0,+),x\longmapsto P(x),$

a group homomorphism?

b) For what real polynomials ${}Q\in \mathbb {R} [X]$ is the polynomial mapping

$(\mathbb {R} ^{\times },1,\cdot )\longrightarrow (\mathbb {R} ^{\times },1,\cdot ),x\longmapsto Q(x),$

well-defined and a group homomorphism?

Exercise

Show that the mapping

$S_{n}\longrightarrow \operatorname {GL} _{n}\!{\left(\mathbb {R} \right)},\pi \longmapsto M_{\pi },$

that sends a permutation ${}\pi$ on ${}\{1,\ldots ,n\}$ to the corresponding permutation matrix ${}M_{\pi }$, is an injective group homomorphism.

Exercise

Let ${}R$ be a commutative ring, and ${}h\in R$. Show that the mapping

$R\longrightarrow R,f\longmapsto hf,$

is a group homomorphism. Describe the image and the kernel of this mapping.

With the concept of residue class formation, the following exercises will become soon easier.

Exercise

Let ${}d\in \mathbb {N} _{+}$, and consider on

${}\mathbb {Z} /(d)=\{0,1,\ldots ,d-1\}\,$

the operation

${}a+b:=(a+b)\mod d={\begin{cases}a+b,{\text{ if }}a+b<d\,,\\a+b-d,{\text{ if }}a+b\geq d\,.\end{cases}}\,$

Show that this defines an associative operation on this set. Moreover, show that this is even a group.

Exercise

Let ${}d\in \mathbb {N} _{\geq 2}$. We consider

${}\mathbb {Z} /(d)=\{0,1,\ldots ,d-1\}\,$

with the addition described in Exercise 44.14 . Show that the mapping

$\psi \colon \mathbb {Z} /(d)\longrightarrow \mathbb {Z} ,r\longmapsto r,$

is not a group homomorphism.

Exercise

Determine the order of every element in the group ${}\mathbb {Z} /(100)$.

Exercise

Let ${}G$ be a group, and ${}h\in G$. Show that the mapping

$G\longrightarrow G,g\longmapsto hgh^{-1},$

is a group automorphism.

Exercise

We consider the mapping

$\operatorname {GL} _{2}\!{\left(\mathbb {R} \right)}\longrightarrow \operatorname {GL} _{2}\!{\left(\mathbb {R} \right)},{\begin{pmatrix}x&y\\z&w\end{pmatrix}}\longmapsto {\begin{pmatrix}8x-4y+14z-7w&-28x+16y-49z+28w\\2x-y+4z-2w&-7x+4y-14z+8w\end{pmatrix}}.$

Show that this is an inner automorphism.

Exercise

Let ${}a,b,c,d$ be real numbers satisfying ${}ad-bc=1$. Show that the mapping

$\operatorname {GL} _{2}\!{\left(\mathbb {R} \right)}\longrightarrow \operatorname {GL} _{2}\!{\left(\mathbb {R} \right)},{\begin{pmatrix}x&y\\z&w\end{pmatrix}}\longmapsto {\begin{pmatrix}adx-acy+bdz-bcw&-abx+a^{2}y-b^{2}z+abw\\cdx-c^{2}y+d^{2}z-cdw&-bcx+acy-bdz+adw\end{pmatrix}}$

is an inner automorphism.

Exercise

Let ${}M$ be a finite set, and ${}T\subseteq M$ a subset. Let ${}\operatorname {Perm} \,(T)$ and ${}\operatorname {Perm} \,(M)$ be the corresponding permutation groups. Show that

$\Psi \colon \operatorname {Perm} \,(T)\longrightarrow \operatorname {Perm} \,(M),\varphi \longmapsto {\tilde {\varphi }},$

defined by

${}{\tilde {\varphi }}(x)={\begin{cases}\varphi (x),\,{\text{if }}x\in T,\\x{\text{ else}},\end{cases}}\,$

is an injective group homomorphism.

Exercise

Let ${}G$ be a group, and ${}g\in G$ be a group element. Let

$\varphi \colon G\longrightarrow G,h\longmapsto hg,$

be the multiplication with ${}g$. Show that ${}\varphi$ is bijective, and that ${}\varphi$ is a group homomorphism if and only if ${}g=e_{G}$.

Exercise

Does there exist a group homomorphism

$(\mathbb {R} ,+,0)\longrightarrow (\mathbb {R} ,+,0)$

that is not ${}\mathbb {R}$-linear?

Hand-in-exercises

Exercise (3 (1+2) marks)

Let ${}G_{1},\ldots ,G_{n}$ be groups.

a) Define a group structure on the product

$G_{1}\times \cdots \times G_{n}.$

b) Let ${}H$ be another group. Show that a mapping

$\varphi \colon H\longrightarrow G_{1}\times \cdots \times G_{n},x\longmapsto \varphi (x)=(\varphi _{1}(x),\ldots ,\varphi _{n}(x)),$

is a group homomorphism if and only if all components ${}\varphi _{i}$ are group homomorphisms.

Exercise (2 marks)

Determine the order of every element in the group ${}\mathbb {Z} /(12)$.

Exercise (3 marks)

Determine the group homomorphisms from ${}(\mathbb {Q} ,+,0)$ to ${}(\mathbb {Z} ,+,0)$.

Exercise (2 marks)

Determine for every ${}n\in \mathbb {N}$ the kernel of taking powers

$\mathbb {R} ^{\times }\longrightarrow \mathbb {R} ^{\times },z\longmapsto z^{n}.$

When is the mapping surjective?

Exercise (1 mark)

Show that there does not exist a group homomorphism

$\varphi \colon (\mathbb {R} ,0,+)\longrightarrow G$

to a group ${}G$ with the property that ${}r\in \mathbb {R}$ is irrational if and only if ${}\varphi (r)=0$.

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