Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 50

Exercises

Exercise

Determine the symmetry group and the proper symmetry group of the standard vectors ${}\{e_{1},e_{2},e_{3}\}$ in ${}\mathbb {R} ^{3}$.

Exercise

Show that for every ${}n\in \mathbb {N}$, there exists a configuration ${}T$ of ${}n$ points in ${}\mathbb {R} ^{3}$ such that the proper symmetry group of ${}T$ is infinite.

Exercise

Let ${}T$ be a configuration consisting of finitely many points in ${}\mathbb {R} ^{3}$, and suppose that they are not all collinear. Show that the proper symmetry group of ${}T$ is finite.

Exercise

Give an example of a set ${}T\subseteq \mathbb {R} ^{3}$ consisting of two elements and such that its proper symmetry group is trivial.

Exercise

Determine the symmetry group and the proper symmetry group of a line ${}G\subset \mathbb {R} ^{3}$ through the origin.

Exercise

Determine the symmetry group and the proper symmetry group of a plane ${}E\subset \mathbb {R} ^{3}$ through the origin.

Exercise

Let

${}S\subseteq T\subseteq \mathbb {R} ^{3}\,$

be subsets with the corresponding proper symmetry groups ${}H$ and ${}G$. Show that, in general, there is no inclusion between these groups.

Exercise

Let ${}A_{n}$ be an alternating group with ${}n\geq 4$. Show that ${}A_{n}$ is not commutative.

Exercise

Let ${}E$ be a regular ${}n$-gon in ${}\mathbb {R} ^{2}$ with the origin as center, and with ${}(1,0)$ as one vertex. Determine the matrices that describe the improper symmetries of ${}E$ with respect to the standard basis.

Exercise

Let ${}G\subseteq \operatorname {O} _{3}\!{\left(\mathbb {R} \right)}$ be a finite group, and ${}H=G\cap \operatorname {SO} _{3}\!{\left(\mathbb {R} \right)}$. Show that either ${}H=G$, or the number of ${}H$ is half of the number of ${}G$.

Exercise

Let ${}n\in \mathbb {N} _{+}$. Define an injective group homomorphism

$\operatorname {O} _{n}\!{\left(\mathbb {R} \right)}\longrightarrow \operatorname {SO} _{n+1}\!{\left(\mathbb {R} \right)},$

that is, from the group of isometries on ${}\mathbb {R} ^{n}$ to the group of proper isometries on ${}\mathbb {R} ^{n+1}$.

The next exercises refer to the so-called Klein four-group. This is simply the product group ${}\mathbb {Z} /(2)\times \mathbb {Z} /(2)$.

Exercise

Show that the Klein four-group is isomorphic to a subgroup of the permutation group ${}S_{4}$. How does a realization as a subgroup of the cube group look like?

Exercise

Show that the dihedral group ${}D_{2}$ is isomorphic to the Klein four-group.

Exercise

Show that the dihedral group ${}D_{3}$ is isomorphic to the permutation group ${}S_{3}$.

Exercise

We consider the double pyramid of height ${}5$ over the square with the vertices ${}(\pm 1,\pm 1)$. What is the name of the proper symmetry group of this object? Determine the matrices that describe, with respect to the standard basis, the proper symmetries of this double pyramid.

Exercise

Show that the dihedral groups ${}D_{n}$, ${}n\geq 3$, are not commutative.

Exercise

Let ${}W$ be the group of the proper symmetries of a cube. Give a chain of successive subgroups (as long as possible)

${}\{\operatorname {id} \}\subset G_{1}\subset G_{2}\subset \ldots \subset G_{n}=W\,$

such that between ${}G_{i}$ and ${}G_{i+1}$ there is no further subgroup.

Exercise

Determine the proper symmetry group of the axes of coordinates in ${}\mathbb {R} ^{3}$.

Exercise

Let ${}G\subseteq \operatorname {SO} _{3}\!{\left(\mathbb {R} \right)}$ be a finite subgroup. Show that the stabilizer group of a semiaxis from the system of semiaxes of ${}G$ is cyclic.

Exercise

Let ${}G\subseteq \operatorname {SO} _{3}\!{\left(\mathbb {R} \right)}$ be the proper symmetry group of a cube with edges parallel to the coordinate axes. Give explicitly (in matrix description) inner automorphisms of the cube group that transform the following stabilizer groups of a semiaxis into the stabilizer group of another semiaxis. What matrices correspond to what matrices?

a) The stabilizer group of the positive ${}x$-axis and the stabilizer group of the positive ${}z$-axis.

b) The stabilizer group of the space diagonal ${}\mathbb {R} _{+}{\begin{pmatrix}1\\1\\1\end{pmatrix}}$ and the stabilizer group of the space diagonal ${}\mathbb {R} _{+}{\begin{pmatrix}1\\1\\-1\end{pmatrix}}$.

c) The stabilizer group of the axis ${}\mathbb {R} _{+}{\begin{pmatrix}1\\1\\0\end{pmatrix}}$ (given by midpoints of opposite edges) and the stabilizer group of ${}\mathbb {R} _{+}{\begin{pmatrix}1\\-1\\0\end{pmatrix}}$.

Exercise

Let ${}{\mathbb {F} }_{q}$ be a finite field (with ${}q$ elements). Determine the number of elements in

$\operatorname {GL} _{n}\!{\left({\mathbb {F} }_{q}\right)}.$

The following exercises use the center of a group.

Let ${}G$ be a group. The center ${}Z=Z(G)$ of ${}G$ is the subset

${}Z={\left\{g\in G\mid gx=xg{\text{ for all }}x\in G\right\}}\,.$

Exercise

Let ${}G$ be a group. Show that the center ${}Z(G)$ is a subgroup of ${}G$.

Exercise

Let ${}G$ be a group. Show that the center ${}Z\subseteq G$ is a normal subgroup in ${}G$. Relate the center to the group homomorphism

$\kappa \colon G\longrightarrow \operatorname {Aut} \,(G),g\longmapsto \kappa _{g}.$

Was is the image of this homomorphism, and what is the content of the homomorphism theorems in this situation?

Exercise

Consider the following thought experiment: given a sphere made of metal, and ${}n$ identical particles with the same positive electric charge. Therefore, these particles repel each other. We put these particles on the sphere, they still repel each other, but they stay on the sphere. What is the (final) configuration of these particles? Don't we expect "due to physical laws“ that there is a "uniform“ configuration where all particles are on an equal footing? Don't we expect that for any two particles ${}P,Q$, there exists a symmetry of the sphere and of the configuration that transforms ${}P$ into ${}Q$?

Hand-in-exercises

Exercise (2 marks)

We consider the action of the tetrahedral group on the four vertices of a tetrahedron. Show that this yields an isomorphism between the tetrahedral group and the alternating group ${}A_{4}$.

Exercise (3 marks)

We consider a regular ${}n$-gon and the corresponding group of all (proper and improper) symmetries, that is, the dihedral group ${}D_{n}$. Describe ${}D_{n}$ as a subgroup of the permutation group ${}S_{n}$. Which permutations generate the dihedral group? For which ${}n$ is the dihedral group a subgroup of the alternating group?

Exercise (2 marks)

Let ${}G\subseteq \operatorname {O} _{2}\!$ be a finite subgroup of the (improper) isometry group of the real plane, and let ${}G\not \subseteq \operatorname {SO} _{2}\!$. Show that there exists a surjective group homomorphism

$G\longrightarrow \mathbb {Z} /(2)$

such that its kernel is a cyclic group. Conclude that the order of ${}G$ is even.

Exercise (3 marks)

Let ${}G$ be a group with center ${}Z(G)$. Prove the following statements:

a) ${}G$ is Abelian if and only if ${}G/Z(G)$ is cyclic.

b) The index of ${}Z(G)$ in ${}G$ is not a prime number.

c) If the order of ${}G$ is ${}pq$ for two prime number ${}p$ and ${}q$, then ${}G$ is Abelian, or ${}Z(G)$ is trivial.

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