Week 9 Problem Sheet
Group Theory and Linear algebra
Semester II 2011

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 1 September 2011

(1) Week 9: Vocabulary
(2) Week 9: Results
(3) Week 9: Examples and computations

Week 9: Vocabulary

	Define a dihedral group and give some illustrative examples.
	Define a rotation in ${ℝ}^2$ and give some illustrative examples.
	Define a rotation in ${ℝ}^3$ 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.

Week 9: Results

	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 $S=\left\{\right(g,x)\in G\times X|g\cdot x=x\}$. 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 $G$.
	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 $G\simeq D_p$.

Week 9: Examples and computations

	Let $H$ denote the subgroup of $D_4=⟨a,b|a^4=1,b^2=1,ba{b}^{-1}=a^{-1}⟩$ 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 $D_4=⟨a,b|a^4=1,b^2=1,ba{b}^{-1}=a^{-1}⟩$ 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 $$U=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),I=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right),J=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),K=\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right).$$ Show that $$I^2=J^2=K^2=-U,IJ=K,JK=I,KI=J,$$ and that $Q_8$ is a subgroup of ${\mathrm{GL}}_2\left(ℂ\right)$.
	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 $D_8=⟨a,b⟩$ 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 $$a^n=1,b^2=1,ba{b}^{-1}=a^{-1}.$$ 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.
	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 ${\mathrm{GL}}_n\left(𝔽\right)$ on ${𝔽}^n$.
	Describe precisely the action of ${\mathrm{GL}}_n\left(𝔽\right)$ on the set of bases of the vector space ${𝔽}^n$ and prove that this action is well defined.
	Describe precisely the action of ${\mathrm{GL}}_n\left(𝔽\right)$ on the set of subspaces of the vector space ${𝔽}^n$ 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 $G={\mathrm{SO}}_2\left(ℝ\right)$ on the set $X={ℝ}^2$.
	Find the orbits and stabilisers for the action of $G={\mathrm{SO}}_3\left(ℝ\right)$ on the set $X={ℝ}^3$.
	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 $G=⟨\left(123\right)⟩$. 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 $G=⟨\left(1234\right)⟩$. 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 $G=⟨\left(12\right),\left(34\right)⟩$. 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 $G=⟨\left(1234\right),\left(13\right)⟩$ (isomorphic to a dihedral group of order 8). Describe the orbits and stabilizers for the action of $G$ on $X$.
	Let $G=ℝ$ (with operation addition) and let $X={ℝ}^3$. Let $v\in {ℝ}^3$. 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 $$(1,12)(3,10)(5,13)(11,15),(2,7)(4,14)(6,10)(9,13),\text{and}(4,8)(6,10)(7,12)(9,11).$$ 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 $𝔽$ be a field and let $n\in {\mathbb{Z}}_{>0}$. Determine the centre of ${\mathrm{GL}}_n\left(𝔽\right)$.
	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 ${\mathrm{GL}}_3\left(ℝ\right)$ of the following matrices: $$\left(\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right)\text{and}\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 2\end{array}\right).$$
	Determine the centralizer in ${\mathrm{GL}}_3\left(ℝ\right)$ of the following matrices: $$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 2\end{array}\right)\text{and}\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 0\\ 0& 0& 2\end{array}\right).$$
	Determine the centralizer in ${\mathrm{GL}}_3\left(ℝ\right)$ of the following matrices: $$\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 0\\ 0& 0& 2\end{array}\right)\text{and}\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right).$$
	Determine the centralizer in ${\mathrm{GL}}_3\left(ℝ\right)$ of the following matrices: $$\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\text{and}\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 0& 0& 1\end{array}\right).$$
	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 ${\mathrm{GL}}_2\left(ℝ\right)$ act on ${ℝ}^2$ in the usual way: $A\cdot \overrightarrow{x}=A\overrightarrow{x}$, for $A\in {\mathrm{GL}}_2\left(ℝ\right)$ and $\overrightarrow{x}$ a column vector in ${ℝ}^2$. 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 $G$.
	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 S4 conjugating $\sigma =\left(1234\right)$ to $\tau \sigma {\tau }^{-1}=\left(2413\right)$. (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 $S_4$.
	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 ${\mathrm{GL}}_2\left(ℝ\right)$ of the matrix $\left(\begin{array}{cc}2& 1\\ 0& 2\end{array}\right)$.
	Show that ${\mathrm{SL}}_2\left(ℝ\right)$ acts on the upper half plane $H=\{z\in ℂ|\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$.$

References

[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.