Group Theory and Linear algebra

Semester II 2011

Last updates: 1 September 2011

(1) Week 10: Vocabulary

(2) Week 10: Results

(3) Week 10: Examples and computations

Define ${\mathbb{R}}^{2}$ and ${\mathbb{E}}^{2}$ and give some illustrative examples. | |

Define isometry of ${\mathbb{E}}^{2}$ and give some illustrative examples. | |

Define a rotation of ${\mathbb{E}}^{2}$ and give some illustrative examples. | |

Define a reflection of ${\mathbb{E}}^{2}$ and give some illustrative examples. | |

Define a translation of ${\mathbb{E}}^{2}$ and give some illustrative examples. | |

Define glide reflection of ${\mathbb{E}}^{2}$ and give some illustrative examples. | |

Define ${\mathbb{R}}^{n}$ and ${\mathbb{E}}^{n}$ and give some illustrative examples. | |

Define isometry of ${\mathbb{E}}^{n}$ and give some illustrative examples. | |

Define a rotation of ${\mathbb{E}}^{n}$ and give some illustrative examples. | |

Define a reflection of ${\mathbb{E}}^{n}$ and give some illustrative examples. | |

Define a translation of ${\mathbb{E}}^{n}$ and give some illustrative examples. | |

Define the groups ${O}_{n}\left(\mathbb{R}\right)$ and ${\mathrm{SO}}_{n}\left(\mathbb{R}\right)$ and give some illustrative examples. | |

Define a rotation in ${\mathbb{R}}^{2}$ and give some illustrative examples. | |

Define a rotation in ${\mathbb{R}}^{3}$ and give some illustrative examples. |

Show that if an isometry fixes two points then it fixes all points of the line on which they lie. | |

Show that if an isometry fixes three points which do not all lie on a line then it fixes all of ${\mathbb{E}}^{2}$. | |

Let ${\sigma}_{1}$ and ${\sigma}_{2}$
be reflections in axes ${L}_{1}$ and
${L}_{2}$. Show that
- (a) If ${L}_{1}$ and ${L}_{2}$ intersect then the product ${\sigma}_{1}{\sigma}_{2}$ is a rotation about the point of intersection of ${L}_{1}$ and ${L}_{2}$ with an angle of rotation twice the angle between ${L}_{1}$ and ${L}_{2}$, and
- (b) If ${L}_{1}$ and ${L}_{2}$ are parallel then the product ${\sigma}_{1}{\sigma}_{2}$ is a translation in a direction perpendicular to ${L}_{i}$ with a magnitude equal to twice the distance between ${L}_{1}$ and ${L}_{2}$.
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Show that the product of three reflections in parallel axes is a reflection. | |

Show that the product of three reflections in axes which are not parallel and which do not intersect in a point is a glide reflection. | |

Show that the set of fixed points of an isometry is one of the following:
- (1) All of ${\mathbb{E}}^{2}$, in which case the isometry is the identity;
- (2) A line in ${\mathbb{E}}^{2}$, in which case the isometry is the reflection in that line;
- (3) A single point, in which case the isometry is a rotation about that point and can be expressed as the product of two reflections;
- (4) empty, in which case the isometry is either (a) a translation and can be expressed as the product of two reflections or (b) a glide reflection and can be expressed as the product of three reflections.
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Let $\mathcal{I}$ be the group of isometries of ${\mathbb{E}}^{2}$. Show that the set of translations forms a normal subgroup of $\mathcal{I}$. | |

Let $\mathcal{I}$ be the group of isometries of ${\mathbb{E}}^{2}$. Let $P$ be a point of ${\mathbb{E}}^{2}$. Show that the set of isometries of ${\mathbb{E}}^{2}$ which fix $P$ is a subgroup of $\mathcal{I}$. | |

Let $\mathcal{I}$ be the group of isometries of ${\mathbb{E}}^{2}$. Let $P$ and $Q$ be points of ${\mathbb{E}}^{2}$. Let ${\mathcal{O}}_{P}$ be the set of isometries that fix $P$ and let ${\mathcal{O}}_{Q}$ be the sets of isometries that fix $Q$. Show that ${\mathcal{O}}_{P}$ and ${\mathcal{O}}_{Q}$ are conjugate subgroups of $\mathcal{I}$. | |

Let $\mathcal{I}$ be the group of isometries of ${\mathbb{E}}^{2}$. Let $P$ be a point of ${\mathbb{E}}^{2}$. Show that every element of $\mathcal{I}$ can be uniquely expressed as a product of a translation and an isometry fixing $P$. | |

Let $\mathcal{I}$ be the group of isometries of ${\mathbb{E}}^{2}$. Let $P$ be a point of ${\mathbb{E}}^{2}$. Let ${\mathcal{O}}_{P}$ be the set of isometries that fix $P$. Show that there is a surjective homomorphism ${\pi}_{P}:\mathcal{I}\to {\mathcal{O}}_{P}$. | |

Show that a finite group of isometries of ${\mathbb{E}}^{2}$ is a cyclic group or a dihedral group. | |

Let $f$ be an isometry of ${\mathbb{E}}^{n}$ such that $f\left(0\right)=0$. Show that there exists an orthogonal matrix $A\in {O}_{n}\left(\mathbb{R}\right)$ such that $f\left(x\right)=Ax$, for $x\in {\mathbb{E}}^{n}$. | |

Show that if $f:{\mathbb{E}}^{n}\to {\mathbb{E}}^{n}$ then there exist $A\in {O}_{n}\left(\mathbb{R}\right)$ and $b\in {\mathbb{R}}^{n}$ such that $f\left(x\right)=Ax+b$. |

Describe the rotational symmetries of a cube. There are 24 in all. Are there any other symmetries besides these rotations? | |

Describe the 12 rotational symmetries of a regular tetrahedron. | |

Find two “different” multiplication tables for groups with 4 elements. Show that both can be represented as symmetry groups of geometric figures in ${\mathbb{R}}^{2}$. | |

Let $A\in {O}_{n}\left(\mathbb{R}\right)$. Show that the linear transformation $f:{\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ defined by $f\left(x\right)=Ax$ is an isometry. | |

Let $b\in {\mathbb{R}}^{n}$. Show that the function ${t}_{b}:{\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ given by ${t}_{b}\left(x\right)=x+b$ is an isometry. Show that the inverse of ${t}_{b}$ is ${t}_{-b}$. | |

Show that compositions of isometries are isometries. | |

Define a "reflection in a line" in ${\mathbb{E}}^{2}$ and show that it is an isometry. | |

Define a "rotation about a point" in ${\mathbb{E}}^{2}$ and show that it is an isometry. | |

Define a "translation" in ${\mathbb{E}}^{2}$ and show that it is an isometry. | |

Define a "glide relfection" in ${\mathbb{E}}^{2}$ and show that it is an isometry. | |

Let $\mathcal{I}$ be the group of isometries of ${\mathbb{E}}^{2}$. Let ${\mathcal{I}}_{+}$ denote the subset of $\mathcal{I}$ consisting of all translations together with all rotations. Show that ${\mathcal{I}}_{+}$ is a subgroup of $\mathcal{I}$. | |

Let $\mathcal{I}$ be the group of isometries of ${\mathbb{E}}^{2}$. Let ${\mathcal{I}}_{+}$ denote the subset of $\mathcal{I}$ consisting of all translations together with all rotations. Show that ${\mathcal{I}}_{+}$ is a subgroup of index 2 in $\mathcal{I}$ and that ${\mathcal{I}}_{+}$ is a normal subgroup of $\mathcal{I}$. | |

Let $\mathcal{I}$ be the group of isometries of ${\mathbb{E}}^{2}$. Let ${\mathcal{I}}_{+}$ denote the subset of $\mathcal{I}$ consisting of all translations together with all rotations. Show that $f\in {\mathcal{I}}_{+}$ if and only if $f$ is a product of an even number of reflections. | |

Identify ${\mathbb{E}}^{2}$ with the complex plane so that each point of ${\mathbb{E}}^{2}$ can be represented by a complex number. Show that every isometry can be represented in the form $z\mapsto {e}^{i\theta}z+u$ or of the form $z\mapsto {e}^{i\theta}\stackrel{\u203e}{z}+u$, for some real number $\theta $ and some complex number $u$. Show that the former type correspond to orientation preserving isometries. | |

Let $\mathcal{I}$ be the group of isometries of ${\mathbb{E}}^{2}$. Describe the conjugacy classes in the group $\mathcal{I}$. | |

Show that if $f$ and $g$ are isometries of ${\mathbb{E}}^{n}$ then so is $f\circ g$. | |

Let $(A,b)$ denote the isometry of
${\mathbb{E}}^{n}$ given by
$x\mapsto Ax+b$
for $A\in O\left(n\right)$,
$b\in {\mathbb{R}}^{n}$.
- (a) Show that the function $\pi :\mathrm{isom}\left({\mathbb{E}}^{n}\right)\to O\left(n\right)$ given by $\pi \left(\right(A,b\left)\right)=A$ is a homomorphism.
- (b) Find the kernel and image of $\pi $.
- (c) Deduce that the set $T$ of all translations is a normal subgroup of $\mathrm{isom}\left({\mathbb{E}}^{n}\right)$ with $\mathrm{isom}\left({\mathbb{E}}^{n}\right)/T$ isomorphic to $O\left(n\right)$.
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Show that the subset ${\mathrm{isom}}_{+}\left({\mathbb{E}}^{n}\right)$ of orientation preserving isometries of ${\mathbb{E}}^{n}$ is a normal subgroup of index 2 in $\mathrm{isom}\left({\mathbb{E}}^{n}\right)$. | |

Write each of the following isometries of ${\mathbb{E}}^{2}$ in the
form $(A,b)$, where
$A\in O\left(2\right)$ and
$b\in {\mathbb{R}}^{2}$.
- (i) $f$ is the anticlockwise rotation through $\pi /2$ about the point $(0,0)$.
- (ii) $g$ is the anticlockwise rotation through $\pi $ about the point $(1,0)$.
- (iii) $h$ is the reflection in the line $x+y+2=0$.
- (iv) $f\circ g$ and $g\circ f$.
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Let $f$ and $g$ be the isometries of ${\mathbb{E}}^{2}$ given by: $f$ is the anticlockwise rotation through $\pi /2$ about the point $(0,0)$ and $g$ is the anticlockwise rotation through $\pi $ about the point $(1,0)$. Show that $f\circ g$ and $g\circ f$ are rotations and find the fixed point and the angle of rotation for each of them. | |

Let ${R}_{1}$ and ${R}_{2}$ be reflections in the lines $y=0$ and $y=a$, respectively. Find formulas for ${R}_{1}$ and ${R}_{2}$ and verify that ${R}_{1}\circ {R}_{2}$ and ${R}_{2}\circ {R}_{1}$ are translations. | |

Let $f$ be an orientation reversing isometry of ${\mathbb{E}}^{2}$. Show that ${f}^{2}$ is a translation. | |

Let $\mathrm{Fix}\left(h\right)=\left\{x\phantom{\rule{0.3em}{0ex}}\right|\phantom{\rule{0.3em}{0ex}}h\left(x\right)=x\}$. Show that if $f:{\mathbb{E}}^{2}\to {\mathbb{E}}^{2}$ and $g:{\mathbb{E}}^{2}\to {\mathbb{E}}^{2}$ are isometries then $\mathrm{Fix}\left(gf{g}^{-1}\right)=g\mathrm{Fix}\left(f\right)$. | |

Let $f:{\mathbb{E}}^{2}\to {\mathbb{E}}^{2}$ and $g:{\mathbb{E}}^{2}\to {\mathbb{E}}^{2}$ be isometries. Show that if $f$ is the reflection in a line $L$ then $gf{g}^{-1}$ is reflection in the line $g\left(L\right)$. | |

Let $f:{\mathbb{E}}^{2}\to {\mathbb{E}}^{2}$ and $g:{\mathbb{E}}^{2}\to {\mathbb{E}}^{2}$ be isometries. Show that if $f$ is a rotation by $\theta $ about $p$ then $gf{g}^{-1}$ is a rotation about $g\left(p\right)$ by $\theta $ if $g$ preserves orientation and by $-\theta $ if $g$ reverses orientation. | |

Let $f:{\mathbb{E}}^{2}\to {\mathbb{E}}^{2}$ and $g:{\mathbb{E}}^{2}\to {\mathbb{E}}^{2}$ be isometries. Show that if $f$ is a translation then $gf{g}^{-1}$ is a translation by the same distance. | |

Let ${D}_{\infty}$ be the set of isometries of ${\mathbb{E}}^{2}$ consisting of all translations by $(n,0)$ and all reflections in the lines $x=n/2$, where $n\in \mathbb{Z}$. Show that ${D}_{\infty}$ is a subgroup of $\mathrm{isom}\left({\mathbb{E}}^{2}\right)$. | |

Let ${D}_{\infty}$ be the set of isometries of ${\mathbb{E}}^{2}$ consisting of all translations by $(n,0)$ and all reflections in the lines $x=n/2$, where $n\in \mathbb{Z}$. Show that ${D}_{\infty}$ acts on the $x$-axis and find the orbit and stabilizer of each of the points $(1,0)$, $(\frac{1}{2},0)$, $(\frac{1}{3},0)$. | |

Let ${D}_{\infty}$ be the set of isometries of ${\mathbb{E}}^{2}$ consisting of all translations by $(n,0)$ and all reflections in the lines $x=n/2$, where $n\in \mathbb{Z}$. Show that ${D}_{\infty}$ is generated by $a:(x,y)\mapsto (x+1,y)$ and $b:(x,y)\mapsto (-x,y)$ and that these satisfy the relations ${b}^{2}=1$ and $ba{b}^{-1}={a}^{-1}$. | |

Show that every orientation preserving isometry of ${\mathbb{E}}^{3}$ is either: (i) a rotation about an axis, (ii) a translation, of (iii) a screw motion consisting of a rotation about an axis composed with a translation parallel to that axis. | |

Show that a rotation fixing the origin on ${\mathbb{R}}^{3}$ has an eigenvalue 1. Show that the corresponding eigenspace is of dimension 1, the axis of rotation. | |

Show that a rotation fixing the origin on ${\mathbb{R}}^{2}$ has two eigenvalues 1 and -1. Show that the eigenspace corresponding to 1 is the line of reflection and that the eigenspace corresponding to -1 is the perpendicular to the line of reflection. | |

Let $f$ be a rotation on ${\mathbb{R}}^{3}$. Then the plane perpendicular to the axis of rotation is an invariant subspace of $f$. Show that the matrix for the rotation with respect to a basis of two orthonormal vectors from the plane and a unit vector along the axis of rotation is $$\left(\begin{array}{ccc}\mathrm{cos}\theta & -\mathrm{sin}\theta & 0\\ \mathrm{sin}\theta & \mathrm{cos}\theta & 0\\ 0& 0& 1\end{array}\right).$$ | |

Let $f$ be a reflection, in a line through the origin, in ${\mathbb{R}}^{2}$. Show that the minimal polynomial of $f$ is ${x}^{2}-1$. | |

Define a 4-dimensional cube and work out some of its rotational symmetries. | |

What letters in the Roman alphabet display symmetry? | |

Show that the set of all rotations of the plane about a fixed centre P, together with the operation of composition of symmetries, form a group. What about all of the reflections for which the axis (or mirror) passes through P? | |

Describe the product of a rotation of the plane with a translation. Describe the product of two (planar) rotations about different axes. | |

Find the order of a reflection. | |

Find the order of a translation in the group of symmetries of a plane pattern. | |

Can you find an example of two symmetries of finite order where the product is of infinite order? | |

Let $G$ be the group of symmetries of a plane tesselation. Decide whether the set of rotations in $G$ is a subgroup. |

[GH]
J.R.J. Groves and
C.D. Hodgson,
*Notes for 620-297: Group Theory and Linear Algebra*, 2009.

[Ra]
A. Ram,
*Notes in abstract algebra*, University of Wisconsin, Madison 1994.