Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 51

Exercises

Exercise

Consider the proof zu Lemma 51.1 with the notation used there. Justify the following statements.

a) A proper isometry with two fixed axes is the identity.

b) $G$ is the union of all $G_H$.

c) Let $g\ne \mathrm{Id}$. The element $g$ appears in exactly two $G_H$. In which?

d) The class of semiaxes $K_i$ contains $n/n_i$ elements.

Exercise

Check the formula

$2(1-\frac{1}{n})=\sum _{i=1}^m(1-\frac{1}{n_i})$

of

Lemma 51.1 for the octahedron, the dodecahedron, and the icosahedron.

Exercise

Let $G\subseteq {\mathrm{SO}}_3\left(ℝ\right)$ be a finite subgroup of the group of proper linear isometries in ${ℝ}^3$. Suppose that there is only one class of semiaxes $K$. What numerical relation would hold between $\mathrm{\#}\left(G\right)$, $\mathrm{\#}\left(K\right)$, and $\mathrm{\#}\left(G_H\right)$ ($H\in K$)? Conclude that such a symmetry group does not exist.

Exercise

Show that the equation

$\frac{2}{n}=\frac{1}{a}+\frac{1}{b}$

in $\mathbb{N}$ has for $a,b\le n$ only the solutions $n=a=b$.

Exercise

Show that the equation

$\frac{2}{n}=\frac{1}{a}+\frac{1}{b}$

in $\mathbb{N}$ also has solutions $a\ne b$.

Exercise

Let $G\subseteq {\mathrm{SO}}_3\left(ℝ\right)$ be a finite subgroup of the group of proper linear isometries in ${ℝ}^3$ with a fixed class of semiaxes $K$. Determine the kernel of the group homomorphism

$G⟶\mathrm{Perm}(K),g⟼{\sigma }_g:H\mapsto g(H).$

Exercise

Let $G\subseteq {\mathrm{SO}}_3\left(ℝ\right)$ be a finite subgroup of the group of proper linear isometries in ${ℝ}^3$ with three classes of semiaxes, and let $K$ denote one of them. Show that the group homomorphism

$G⟶\mathrm{Perm}(K),g⟼{\sigma }_g:H\mapsto g(H),$

is injective. Show that this is not true when there are only to classes of semiaxes.

Exercise

Determine the angles between the semiaxes of (the symmetry group of) the platonic solids.

Exercise

Let two semiaxes $H_1$ and $H_2$ in ${ℝ}^3$ be given. Determine the set of rotation axes and the rotation angles that transform $H_1$ into $H_2$.

Exercise

We consider an equilateral triangle in the $x,y$-plane, with $(0,0)$ as center and with $(1,0)$ as one vertex. Let $D$ denote the bipyramid over this triangle with upper top $(0,0,2)$ and lower top $(0,0,-2)$.

a) Determine the matrices and the rotation axes of the (proper) symmetries that transform $D$ to itself.

b) Determine an operation table for these symmetries.

c) Describe what happens to the three vertices of the triangle under these symmetries.

Exercise

We consider the cube.

Let $\alpha$ be the rotation of the cube around the axis given by the vertices $A$ and $G$ that sends the vertex $B$ to $D$. Let $\beta$ be the half rotation around the vertical axis (that is, the line connecting the center of the face $A,B,C,D$ and the center of the face $E,F,G,H$).

a) Establish the value tables for the permutations on the set of vertices $\{A,B,C,D,E,F,G,H\}$ induced by $\alpha ,\beta ,\alpha \beta$, and $\beta \alpha$.

b) Determine the rotation axis of $\alpha \beta$, and of $\beta \alpha$. Determine also the orders of these rotations.

c) Determine the cycle representation of the permutation on the set of vertices induced by ${\alpha }^2$. What is ${\alpha }^{1001}$?

d) We consider the permutation $\sigma$ that is given by the value table

$x$	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$
$\sigma (x)$	$B$	$C$	$D$	$A$	$G$	$H$	$E$	$F$

Does there exist a rotation of the cube that induces this permutation? Compute the sign of $\sigma$.

Exercise

Let $G$ be a group, let $M$ denote a set, and let

$G⟶\mathrm{Perm}(M),g⟼{\sigma }_g,$

denote a group homomorphism into the permutation group of $M$. Show that this induces in a natural way a group homomorphism

$G⟶\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{(}𝔓\mathrm{(}\mathrm{M}\mathrm{)}\mathrm{)},g⟼(N\mapsto g(N)),$

in the permutation group of the power set.

Exercise

a) Show that the group $\mathbb{Z}/(2)\times \mathbb{Z}/(2)\times \mathbb{Z}/(2)$ is not the proper symmetry group of a subset $T\subseteq {ℝ}^3$.

b) Show that the group $\mathbb{Z}/(2)\times \mathbb{Z}/(2)\times \mathbb{Z}/(2)$ can be realized as a subgroup of the full isometry group ${\mathrm{O}}_3\left(ℝ\right)$.

c)

Consider the proper symmetry group of a rectangular cuboid with three different side lengths. For every axis given by the centers of two opposite faces, the half rotation around this axis is symmetry. Does this contradict part (a)?

Exercise

Show that every finite group can be realized as a subgroup of ${\mathrm{SO}}_n\left(ℝ\right)$.

Hand-in-exercises

Exercise (4 marks)

Let $A_1,A_2,A_3$ and $A_4$ be four lines in ${ℝ}^3$ through the origin fulfilling the property that no three of them lie in a plane. Let

$f:{ℝ}^3⟶{ℝ}^3$

be a linear, proper isometry with $f(A_i)=A_i$ for $i=1,2,3,4$. Show that $f$ is the identity. Give an example to show that this statement is not true without the plane condition.

Exercise (6 marks)

Let ${\phi }_1,{\phi }_2,{\phi }_3$ be rotations around the $x$-axis, the $y$-axis, and the $z$-axis, with orders ${\ell }_1,{\ell }_2,{\ell }_3$ (that is, ${\phi }_1$ is a rotation about the angle $360/{\ell }_1$ degree around the $x$-axis, etc.). Let $1\le {\ell }_1\le {\ell }_2\le {\ell }_3$. For which tuples $({\ell }_1,{\ell }_2,{\ell }_3)$ is the group generated by these three rotations finite?

Hint: Exercise 51.8 .

Exercise (3 marks)

Show that the alternating groups $A_n$ do not contain a subgroup of index two.

Hint: Exercise 18.18 .

Exercise (4 marks)

Show that the group $\mathbb{Z}/(2)\times \mathbb{Z}/(4)$ is not the proper symmetry group of a subset $T\subseteq {ℝ}^3$.

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