Group Theory and Linear algebra

Semester II 2011

Last updates: 1 September 2011

(1) Week 9: Vocabulary

(2) Week 9: Results

(3) Week 9: Examples and computations

Define a dihedral group 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. | |

Define a $G$-action on $X$ and give some illustrative examples. | |

Define a $G$-set and give some illustrative examples. | |

Define orbits and stabilizers and give some illustrative examples. | |

Define the action of $G$ on itself by left multiplication and the action of $G$ on itself by conjugation and give some illustrative examples. | |

Define conjugate, conjugacy class, and centralizer and give some illustrative examples. | |

Define the centre of a group and give some illustrative examples. |

Let $G$ be a group and let $X$ be a $G$-set. Let $x\in X$. Show that the stabilizer of $x$ is a subgroup of $G$. | |

Let $G$ be a group and let $X$ be a $G$-set. Show that the orbits partition $G$. | |

Let $G$ be a group and let $X$ be a $G$-set. Let $x\in X$ and let $H$ be the stabilizer of $x$. Show that $\mathrm{Card}\left(G/H\right)=\mathrm{Card}\left(Gx\right)$ and that $$\mathrm{Card}\left(G\right)=\mathrm{Card}\left(Gx\right)\mathrm{Card}\left(H\right).$$ | |

Let $G$ be a group. Show that $G$ is isomorphic to a subgroup of a permutation group. | |

Let $G$ be a finite group acting on a finite set $X$.
For each $g\in G$ let
$\mathrm{Fix}\left(g\right)$ be the set of elements of $X$
fixed by $g$.
- (a) Let $S=\left\{\right(g,x)\in G\times X\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}g\cdot x=x\}$. By counting $S$ in two ways, show that $$\left|S\right|=\sum _{x\in X}\left|\mathrm{Stab}\right(x\left)\right|=\sum _{g\in G}\left|\mathrm{Fix}\right(g\left)\right|.$$
- (b) Show that if $g\cdot x=y$ then $g\mathrm{Stab}\left(x\right){g}^{-1}=\mathrm{Stab}\left(y\right)$, hence $\left|\mathrm{Stab}\right(x\left)\right|=\left|\mathrm{Stab}\right(y\left)\right|$.
- (c) Prove that the number of distinct orbits is $$\frac{1}{\left|G\right|}\sum _{g\in G}\left|\mathrm{Fix}\right(g\left)\right|,$$ i.e. the average number of points fixed by elements of $G$.
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Let $G$ be a finite group. Show that the number of elements of a conjugacy class is equal to the number of cosets of the centralizer of any element of the conjugacy class. | |

Show that the centre of a group $G$ is a normal subgroup of $G$. | |

Let $p$ be a prime, let $n\in {\mathbb{Z}}_{>0}$ and let $G$ be a group of order ${p}^{n}$. Show that $Z\left(G\right)\ne \left\{1\right\}$. | |

Let $p$ be a prime and let $G$ be a group of order ${p}^{2}$. Show that $G$ is isomorphic to $\mathbb{Z}/{p}^{2}\mathbb{Z}$ or $\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p\mathbb{Z}$. | |

Let $G$ be a finite group of order divisible by a prime $p$. Show that $G$ has an element of order $p$. | |

Let $p$ be an odd prime and let $G$ be a group of order $2p$. Show that $G\simeq \mathbb{Z}/2p\mathbb{Z}$ or $G\simeq {D}_{p}$. |

Let $H$ denote the subgroup of ${D}_{4}=\u27e8a,b\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}{a}^{4}=1,{b}^{2}=1,ba{b}^{-1}={a}^{-1}\u27e9$ generated by $a$. Show that $H$ is a normal subgroup of ${D}_{4}$ and write out the multiplication table of ${D}_{4}/H$. | |

Let $H$ denote the subgroup of ${D}_{4}=\u27e8a,b\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}{a}^{4}=1,{b}^{2}=1,ba{b}^{-1}={a}^{-1}\u27e9$ generated by ${a}^{2}$. Show that $H$ is a normal subgroup of ${D}_{4}$ and write out the multiplication table of ${D}_{4}/H$. | |

Find all of the normal subgroups of ${D}_{4}$. | |

The quaternion group is the set
${Q}_{8}=\{\pm U,\pm I,\pm J,\pm K\}$ where
$$U=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),\phantom{\rule{2em}{0ex}}I=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right),\phantom{\rule{2em}{0ex}}J=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),\phantom{\rule{2em}{0ex}}K=\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right).$$
Show that
$${I}^{2}={J}^{2}={K}^{2}=-U,\phantom{\rule{1em}{0ex}}IJ=K,\phantom{\rule{1em}{0ex}}JK=I,\phantom{\rule{1em}{0ex}}KI=J,$$
and that
${Q}_{8}$ is a subgroup
of ${\mathrm{GL}}_{2}\left(\u2102\right)$.
| |

Find all of the cyclic subgroups of the quaternion group ${Q}_{8}$. | |

Show that every subgroup of the quaternion group ${Q}_{8}$, except ${Q}_{8}$ itself, is cyclic. | |

Determine whether ${Q}_{8}$ and ${D}_{4}$ are isomorphic. | |

Let $H$ denote the subgroup of ${D}_{8}=\u27e8a,b\u27e9$ generated by ${a}^{4}$. Write out the multiplication table of ${D}_{8}/H$. | |

Show that the set of rotations in the dihedral group ${D}_{n}$ is a subgroup of ${D}_{n}$. | |

Show that the set of reflections in the dihedral group ${D}_{n}$ is not a subgroup of ${D}_{n}$. | |

Let $n\in {\mathbb{Z}}_{>0}$. Calculate the order of ${D}_{n}$. Always justify your answers. | |

Calculate the orders of the elements of ${D}_{6}$. Always justify your answers. | |

Show that ${D}_{3}$ is isomorphic to ${S}_{3}$. | |

Show that ${D}_{3}$ is nonabelian and noncyclic. | |

Prove that ${D}_{2}$ and $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ are isomorphic. | |

Let $n\in {\mathbb{Z}}_{>0}$. Determine the orders of the elements in the dihedral group ${D}_{n}$. | |

Let $m,n\in {\mathbb{Z}}_{>0}$ such that $m<n$. Show that ${D}_{m}$ is isomorphic to a subgroup of ${D}_{n}$. | |

Determine if the group of symmetries of a rectangle is a cyclic group. | |

Show that the group $\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ and the group ${D}_{4}$ are not isomorphic. | |

Determine all subgroups of the dihedral group ${D}_{5}$. | |

Let $n\in {\mathbb{Z}}_{>0}$. Let $G={D}_{n}$ and $H={C}_{n}$. Compute the cosets of $H$ in $G$ and the index $|G:H|$. | |

Let ${D}_{n}$ be the group of symmetries of a regular $n$-gon. Let $a$ denote a rotation through $2\pi /n$ and let $b$ denote a reflection. Show that $${a}^{n}=1,\phantom{\rule{2em}{0ex}}{b}^{2}=1,\phantom{\rule{2em}{0ex}}ba{b}^{-1}={a}^{-1}.$$ Show that every element of ${D}_{n}$ has a unique expression of the form ${a}^{i}$ or ${a}^{i}b$, where $i\in \{0,1,\dots ,n-1\}$. | |

Determine all subgroups of the dihedral group ${D}_{4}$ as
follows:
- (a) Find all the cyclic subgroups of ${D}_{4}$ by considering the subgroup generated by each element.
- (b) Find two non-cyclic subgroups of ${D}_{4}$.
- (c) Explain why any non-cyclic subgroup of ${D}_{4}$, other than ${D}_{4}$ itself, must be of order 4 and, in fact, must be one of the two subgroups you have listed in the previous part.
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Let $G$ be the group of rotational symmetries of a regular tetrahedron so that $\left|G\right|=12$. Show that $G$ has subgroups of order 1, 2, 3, 4 and 12. | |

Describe precisely the action of ${S}_{n}$ on $\{1,2,\dots ,n\}$ and the action of ${\mathrm{GL}}_{n}\left(\mathbb{F}\right)$ on ${\mathbb{F}}^{n}$. | |

Describe precisely the action of ${\mathrm{GL}}_{n}\left(\mathbb{F}\right)$ on the set of bases of the vector space ${\mathbb{F}}^{n}$ and prove that this action is well defined. | |

Describe precisely the action of ${\mathrm{GL}}_{n}\left(\mathbb{F}\right)$ on the set of subspaces of the vector space ${\mathbb{F}}^{n}$ and prove that this action is well defined. | |

Find the orbits and stabilisers for the action of ${S}_{3}$ on the set $\{1,2,3\}$. | |

Find the orbits and stabilisers for the action of $G={\mathrm{SO}}_{2}\left(\mathbb{R}\right)$ on the set $X={\mathbb{R}}^{2}$. | |

Find the orbits and stabilisers for the action of $G={\mathrm{SO}}_{3}\left(\mathbb{R}\right)$ on the set $X={\mathbb{R}}^{3}$. | |

The dihedral group ${D}_{6}$ acts on a regular hexagon. Colour two opposite sides blue and the other four sides red and let $G$ be the subgroup of ${D}_{6}$ which preserves the colours. Let $X=\{A,B,C,D,E,F\}$ be the set of vertices of the hexagon. Determine the stabilizers and orbits for the action of $G$ on $X$. | |

Since ${S}_{4}$ acts on $X=\{1,2,3,4\}$ any subgroup $G$ acts on $X=\{1,2,3,4\}$. Let $G=\u27e8\left(123\right)\u27e9$. Describe the orbits and stabilizers for the action of $G$ on $X$. | |

Since ${S}_{4}$ acts on $X=\{1,2,3,4\}$ any subgroup $G$ acts on $X=\{1,2,3,4\}$. Let $G=\u27e8\left(1234\right)\u27e9$. Describe the orbits and stabilizers for the action of $G$ on $X$. | |

Since ${S}_{4}$ acts on $X=\{1,2,3,4\}$ any subgroup $G$ acts on $X=\{1,2,3,4\}$. Let $G=\u27e8\left(12\right),\left(34\right)\u27e9$. Describe the orbits and stabilizers for the action of $G$ on $X$. | |

Since ${S}_{4}$ acts on $X=\{1,2,3,4\}$ any subgroup $G$ acts on $X=\{1,2,3,4\}$. Let $G={S}_{4}$. Describe the orbits and stabilizers for the action of $G$ on $X$. | |

Since ${S}_{4}$ acts on $X=\{1,2,3,4\}$ any subgroup $G$ acts on $X=\{1,2,3,4\}$. Let $G=\u27e8\left(1234\right),\left(13\right)\u27e9$ (isomorphic to a dihedral group of order 8). Describe the orbits and stabilizers for the action of $G$ on $X$. | |

Let $G=\mathbb{R}$ (with operation addition) and let $X={\mathbb{R}}^{3}$. Let $v\in {\mathbb{R}}^{3}$. Show that $$\alpha \cdot x=x+\alpha v,$$ defines an action of $G$ on $X$ and give a geometric description of the orbits. | |

Let $G$ be the subgroup of ${S}_{15}$ generated by the three permutations $$(1,12)(3,10)(5,13)(11,15),\phantom{\rule{2em}{0ex}}(2,7)(4,14)(6,10)(9,13),\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}(4,8)(6,10)(7,12)(9,11).$$ Find the orbits of $G$ acting on $X=\{1,2,\dots ,15\}$ and prove that $G$ has order which is a multiple of 60. | |

Let $G$ be a group of order $5$ acting on a set $X$ with 11 elements. Determine whether the action of $G$ on $X$ has a fixed point. | |

Let $G$ be a group of order $15$ acting on a set $X$ with 8 elements. Determine whether the action of $G$ on $X$ has a fixed point. | |

Give an explicit isomorphism between ${D}_{2}$ and a subgroup of ${S}_{4}$. | |

Find the conjugacy classes of ${D}_{4}$. | |

Find the centre of ${D}_{4}$. | |

Let $G$ be a group. Show that $\left\{1\right\}\subseteq Z\left(G\right)$. | |

Show that $Z\left({S}_{3}\right)=\left\{1\right\}$. | |

Let $\mathbb{F}$ be a field and let $n\in {\mathbb{Z}}_{>0}$. Determine the centre of ${\mathrm{GL}}_{n}\left(\mathbb{F}\right)$. | |

Find the conjugacy classes in the quaternion group. | |

Find the conjugates of (123) in ${S}_{3}$ and find the conjugates of (123) in ${S}_{4}$. | |

Find the conjugates of (1234) in ${S}_{4}$ and find the conjugates of (1234) in ${S}_{n}$, for $n\ge 4$. | |

Find the conjugates of $\left(12\dots m\right)$ in ${S}_{n}$, for $n\ge m$. | |

Describe the conjugacy classes in the symmetric group ${S}_{n}$. | |

Suppose that $g$ and $h$ are conjugate elements of a group $G$. Show that ${C}_{G}\left(g\right)$ and ${C}_{G}\left(h\right)$ are conugate subgroups of $G$. | |

Determine the centralizer in ${\mathrm{GL}}_{3}\left(\mathbb{R}\right)$ of the following matrices: $$\left(\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right)\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 2\end{array}\right).$$ | |

Determine the centralizer in ${\mathrm{GL}}_{3}\left(\mathbb{R}\right)$ of the following matrices: $$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 2\end{array}\right)\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 0\\ 0& 0& 2\end{array}\right).$$ | |

Determine the centralizer in ${\mathrm{GL}}_{3}\left(\mathbb{R}\right)$ of the following matrices: $$\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 0\\ 0& 0& 2\end{array}\right)\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right).$$ | |

Determine the centralizer in ${\mathrm{GL}}_{3}\left(\mathbb{R}\right)$ of the following matrices: $$\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 0& 0& 1\end{array}\right).$$ | |

Let $G$ be a group and assume that $G/Z\left(G\right)$ is a cyclic group. Show that $G$ is abelian. | |

Describe the finite groups with exactly one conjugacy class. | |

Describe the finite groups with exactly two conjugacy classes. | |

Describe the finite groups with exactly three conjugacy classes. | |

Let $p$ be a prime. Show that a group of order ${p}^{2}$ is abelian. | |

Let $p$ be a prime and let $G$ be a group of order ${p}^{2}$. Show that $G\simeq Z/p\mathbb{Z}\times Z/p\mathbb{Z}$ or $G\simeq Z/{p}^{2}\mathbb{Z}$. | |

Let $p$ be a prime and let $G$ be a group of order $2p$. Show that $G$ has a subgroup of order $p$ and that this subgroup is a normal subgroup. | |

Let $p$ be a prime. Show that, up to isomorphism, there are exactly two groups of order $2p$. | |

Prove that every nonabelian group of order 8 is isomorphic to the dihedral group ${D}_{4}$ or to the quaternion group ${Q}_{8}$. | |

Show that each group $G$ acts on $X=G$ by right multiplication: $g\cdot x=x{g}^{-1}$, for $g\in G,x\in X$. | |

Let $G={D}_{2}$ act as symmetries of a rectangle. Determine the stabilizer and orbit of a vertex, and the stabilizer and orbit of the midpoint of an edge. | |

Let ${\mathrm{GL}}_{2}\left(\mathbb{R}\right)$ act on ${\mathbb{R}}^{2}$ in the usual way: $A\cdot \overrightarrow{x}=A\overrightarrow{x}$, for $A\in {\mathrm{GL}}_{2}\left(\mathbb{R}\right)$ and $\overrightarrow{x}$ a column vector in ${\mathbb{R}}^{2}$. Determine the stabilizer and orbit of $(0,0)$ and the stabilizer and orbit of $(1,0)$. | |

Let $G$ be the group of rotational symmetries of a regular tetrahedron
$T$.
- (a) For the action of $G$ on $T$, describe the stabilizer and orbit of a vertex, and describe the stabilizer and orbit of the midpoint of an edge.
- (b) Use the results of (a) to calculate the order of $G$ in two different ways.
- (c) By considering the action of $G$ on the set of vertices of $T$, find a subgroup of ${S}_{4}$ isomorphic to $G$.
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A group $G$ of order 9 acts on a set $X$ with 16 elements. Show that there must be at least one point in $X$ fixed by all elements of $G$ (i.e. an orbit consisting of a single element). | |

Find the conjugacy class and centralizer of (12) and (123) in ${S}_{3}$. Check that $\left|\text{conjugacy class}\right|\cdot \left|\text{centralizer}\right|=\left|{S}_{3}\right|$ in each case. | |

Let $\tau $ be a permutation in ${S}_{m}$.
- (a) Let $\sigma $ be an $n$-cycle $\sigma =({a}_{1}{a}_{2}\cdots {a}_{n})$ in ${S}_{m}$. Show that $\tau \sigma {\tau}^{-1}$ takes $\tau \left({a}_{1}\right)\mapsto \tau \left({a}_{2}\right)$, $\tau \left({a}_{2}\right)\mapsto \tau \left({a}_{3}\right)$, ..., $\tau \left({a}_{n}\right)\mapsto \tau \left({a}_{1}\right)$. Hence $\tau \sigma {\tau}^{-1}$ is the $n$-cycle $\left(\tau \right({a}_{1}\left)\tau \right({a}_{2})\cdots \tau ({a}_{n}\left)\right)$.
- (b) Use the previous result to find all conjugates of (123) in ${S}_{4}$.
- (c)
Find a permutation $\tau $ in
conjugating $\sigma =\left(1234\right)$ to $\tau \sigma {\tau}^{-1}=\left(2413\right)$.S 4 - (d) If $\sigma ={\sigma}_{1}\cdots {\sigma}_{k}$, show that $\tau \sigma {\tau}^{-1}=\tau {\sigma}_{1}{\tau}^{-1}\cdots \tau {\sigma}_{k}{\tau}^{-1}$.
- (e) Use the previous results to find all conjugates of $\left(12\right)\left(34\right)$ in ${S}_{4}$.
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Find the number of conjugacy classes in each of ${S}_{3}$, ${S}_{4}$ and ${S}_{5}$ and write down a representative from each conjugacy class. How many elements are in each conjugacy class? | |

Let $H$ be a subgroup of $G$. Show that $H$ is a normal subgroup of $G$ if and only if $H$ is a union of conjugacy classes in $G$. | |

Find normal subgroups of ${S}_{4}$ of order 4 and of order 12. | |

Find the centralizer in ${\mathrm{GL}}_{2}\left(\mathbb{R}\right)$ of the matrix $\left(\begin{array}{cc}2& 1\\ 0& 2\end{array}\right)$. | |

Show that ${\mathrm{SL}}_{2}\left(\mathbb{R}\right)$ acts on the upper half plane $H=\{z\in \u2102\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}\mathrm{Im}z>0\}by$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\cdot z=\frac{az+b}{cz+d}.$$Prove\; that\; this\; action\; is\; well\; defined\; and\; describe\; the\; orbit\; and\; stabiliser\; of$ i$.$ | |

[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.