Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 46

Exercises

Exercise

Determine the cosets of the following subgroups of commutative groups.

a) ${}(\mathbb {Z} ,0,+)\subseteq (\mathbb {R} ,0,+)$.

b) ${}(\mathbb {Q} ,0,+)\subseteq (\mathbb {R} ,0,+)$.

c) ${}(\mathbb {R} ,0,+)\subseteq (\mathbb {C} ,0,+)$.

d) ${}(\mathbb {Z} d,0,+)\subseteq (\mathbb {Z} ,0,+)$ (for ${}d\in \mathbb {N}$).

e) ${}({\left\{z\in \mathbb {C} \mid \vert {z}\vert =1\right\}},1,\cdot )\subseteq (\mathbb {C} \setminus \{0\},1,\cdot )$.

f) ${}({\left\{z\in \mathbb {C} \mid z^{n}=1\right\}},1,\cdot )\subseteq ({\left\{z\in \mathbb {C} \mid \vert {z}\vert =1\right\}},1,\cdot )$ (for ${}n\in \mathbb {N}$).

When do the cosets consist of finitely many elements, when is the index finite?

Exercise

We consider the permutation group ${}S_{4}$, together with the subgroup ${}H$, that is generated by the ${}3$-cycle ${}a\mapsto b,\,b\mapsto c,\,c\mapsto a$. Determine the left cosets and the right cosets of this subgroup.

Exercise

We consider the group ${}G$ of invertible ${}2\times 2$-matrices over the field ${}\mathbb {Z} /(3)$ with ${}3$ elements, and the subgroup ${}H$ of all invertible matrices with determinant ${}1$. Which of the following matrices are equivalent to each other (with respect to ${}H$), and which are not?

${\begin{pmatrix}2&0\\1&2\end{pmatrix}},\,{\begin{pmatrix}0&1\\2&2\end{pmatrix}},\,{\begin{pmatrix}1&1\\2&1\end{pmatrix}},\,{\begin{pmatrix}2&0\\0&2\end{pmatrix}}.$

Exercise

Let ${}G$ be a group, and let ${}H_{1},H_{2}\subseteq H$ denote subgroups with the corresponding equivalence relations ${}\sim _{1}$ and ${}\sim _{2}$ on ${}G$. Show that the equivalence relation to ${}H=H_{1}\cap H_{2}$ is the intersection of the two equivalence relations.

Exercise

Let ${}p$ be a prime number, and let ${}G$ denote a group of order ${}p$. Show that ${}G$ is a cyclic group.

Exercise

Let ${}G$ be s cyclic group. Show that every subgroup of ${}G$ is also cyclic.

Exercise

Let ${}G$ be a finite group. Show that every element ${}g\in G$ has finite order, and show that the powers

$g^{0}=e_{G},\,g^{1}=g,\,g^{2},\ldots ,g^{\operatorname {ord} \,(g)-1}$

are all different.

Exercise

Let ${}R$ be a commutative ring with ${}p$ elements, where ${}p$ is a prime number. Show that ${}R$ is a field.

Exercise

Determine the subgroups of ${}\mathbb {Z} /(15)$.

Exercise

Let ${}G=S_{3}$ be the permutation group of a set with three elements. What numbers appear as an order of a subgroup, and what numbers appear as an order of an element of ${}G$?

Exercise

Let ${}K$ be a field and ${}n\in \mathbb {N} _{+}$. Let ${}\operatorname {GL} _{n}\!{\left(K\right)}$ denote the general linear group of invertible matrices, and

${}\operatorname {SL} _{n}\!{\left(K\right)}\subseteq \operatorname {GL} _{n}\!{\left(K\right)}\,$

the subgroup of matrices with determinant ${}1$. Show that the left coset (and also the right coset) of ${}M\in \operatorname {GL} _{n}\!{\left(K\right)}$ equals the set of all matrices whose determinant coincides with ${}\det M$.

Show in many different ways that ${}\operatorname {SL} _{n}\!{\left(K\right)}$ is a normal subgroup in ${}\operatorname {GL} _{n}\!{\left(K\right)}$.

Exercise

Let ${}G$ and ${}H$ be groups, and let

$\varphi \colon G\longrightarrow H$

be a group homomorphism. Show that the preimage ${}\varphi ^{-1}(N)$ of a normal subgroup ${}N\subseteq H$ is a normal subgroup in ${}G$.

Exercise

Show that the intersection of normal subgroups ${}N_{i}$, ${}i\in I$, in a group ${}G$ is a normal subgroup.

Exercise

Let ${}G$ and ${}H$ be groups, and let

$\varphi \colon G\longrightarrow H$

be a group homomorphism. Is the image of ${}\varphi$ a normal subgroup in ${}H$?

Exercise

Let ${}n\in \mathbb {N} _{+}$. Show that the group of the ${}n$-th roots of unity in ${}\mathbb {C}$ and the group ${}\mathbb {Z} /(n)$ are isomorphic

The next exercise uses the concept of an exact sequence.

Let ${}G_{0},\ldots ,G_{n}$ be groups, and ${}f_{i}\colon G_{i-1}\rightarrow G_{i}$ be group homomorphisms such that ${}\operatorname {ker} {\left(f_{i+1}\right)}=\operatorname {Im} f_{i}$ holds for ${}i=1,\ldots ,n$. In this situation,

$G_{0}\to G_{1}\to \ldots \to G_{n-1}\to G_{n}$

is called an exact sequence of groups.

Exercise

Let

$G_{0}\to G_{1}\to \ldots \to G_{n-1}\to G_{n}$

be an exact sequence of groups, where all groups are finite, and where ${}G_{0}=G_{n}$ are the trivial group. Show that

${}\prod _{i=0}^{n}{\#\left(G_{i}\right)}^{(-1)^{i}}=1\,$

holds.

Hand-in-exercises

Exercise (2 marks)

Determine the subgroups of ${}\mathbb {Z} /(20)$.

Exercise (2 marks)

Determine all groups with four elements.

Exercise (3 marks)

Let ${}M$ be a finite set, and let ${}\sigma$ denote a permutation on ${}M$, and ${}x\in M$. Show that ${}{\left\{n\in \mathbb {Z} \mid \sigma ^{n}(x)=x\right\}}$ is a subgroup of ${}\mathbb {Z}$. We denote the uniquely determined nonnegative generator of this group by ${}\operatorname {ord} _{x}\sigma$. Show the relation

${}\operatorname {ord} \,(\sigma )=\operatorname {lcm} {\left\{\operatorname {ord} _{x}\sigma \mid x\in M\right\}}\,.$

Exercise (2 marks)

Let ${}G$ and ${}H$ be groups, and let

$\varphi \colon G\longrightarrow H$

be a surjective group homomorphism. Show that the image ${}\varphi (N)$ of a normal subgroup ${}N\subseteq G$ is a normal subgroup in ${}H$.

Exercise (2 marks)

Show that every subgroup of index two in a group ${}G$ is a normal subgroup in ${}G$.

Exercise (2 marks)

Let ${}G$ be a group, and let ${}M$ be a set together with an operation. Let

$\varphi \colon G\longrightarrow M$

be a surjective mapping satisfying ${}\varphi (gh)=\varphi (g)\varphi (h)$ for all ${}g,h\in G$. Show that ${}M$ is a group, and that ${}\varphi$ is a group homomorphism.

Exercise (5 marks)

Give an example of subgroups ${}F\subseteq G\subseteq H$ such that ${}F$ is a normal subgroup in ${}G$, and ${}G$ is a normal subgroup in ${}H$, but ${}F$ is not a normal subgroup in ${}H$.

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