Week 8 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 8: Vocabulary
(2) Week 8: Results
(3) Week 8: Examples and computations

Week 8: Vocabulary

	Let $G$ be a group and let $H$ be a subgroup. Define a left coset of $H$, a right coset of $H$ and the index of $H$ in $G$ and give some illustrative examples.
	Let $G$ be a group and let $H$ be a subgroup. Define $G/H$ and give some illustrative examples.
	Let $G$ be a group. Define normal subgroup of $G$ and give some illustrative examples.
	Let $G$ be a group and let $H$ be a normal subgroup. Define the quotient group $G/H$ and give some illustrative examples.

Week 8: Results

	Let $G$ be a group and let $H$ be a subgroup of $G$. Let $a,b\in G$. Show that $Ha=Hb$ if and only if $a{b}^{-1}\in H$.
	Let $G$ be a group and let $H$ be a subgroup of $G$. Show that each element of $G$ lies in exactly one coset of $G$.
	Let $G$ be a group and let $H$ be a subgroup of $G$. Let $a,b\in G$. Show that the function $f:Ha\to Hb$ given by $f\left(ha\right)=hb$ is a bijection.
	Let $G$ be a group and let $H$ be a subgroup of $G$. Show that $G/H$ is a partition of $G$.
	Let $G$ be a group and let $H$ be a subgroup of $G$. Let $g\in G$. Show that $gH$ and $H$ have the same number of elements.
	Let $G$ be a group of finite order and let $H$ be a subgroup of $G$. Show that $\mathrm{Card}\left(H\right)$ divides $\mathrm{Card}\left(G\right)$.
	Let $G$ be a group of finite order and let $g\in G$. Show that the order of $g$ divides the order of $G$.
	Let $G$ be a finite group and let $n=\mathrm{Card}\left(G\right)$. Show that if $g\in G$ then $g^n=1$.
	Let $p$ be a prime positive integer. Show that if $a$ is an integer which is not a multiple of $p$ then $a^{p-1}=1$ mod $p$.
	Let $p$ be a prime positive integer. Let $G$ be a group of order $p$. Show that $G$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$.
	Let $G$ be a group and let $H$ be a subgroup of $G$. Show that $H$ is a normal subgroup of $G$ if and only if $H$ satisfies $$\text{if}g\in G\text{then}Hg=gH.$$
	Let $G$ be a group and let $H$ be a subgroup of $G$. Show that $H$ is a normal subgroup of $G$ if and only if $H$ satisfies $$\text{if}g\in G\text{then}gH{g}^{-1}=H.$$
	Let $G$ be a group and let $H$ be a normal subgroup of $G$. Show that if $a,b\in G$ then $HaHb=Hab$.
	Let $G$ be a group and let $H$ be a normal subgroup of $G$. Show that $G/H$ with operation given by $\left(g_{1}H\right)\left(g_{2}H\right)=g_1{g}_{2}H$ is a group.
	Let $f:G\to H$ be a group homomorphism. Show that $\ker f$ is a normal subgroup of $G$.
	Let $f:G\to H$ be a group homomorphism. Show that $\mathrm{im}f$ is a subgroup of $H$.
	Let $f:G\to H$ be a group homomorphism. Show that $f$ is injective if and only if $\ker f=\left\{1\right\}$.
	Let $G$ be a group and let $H$ be a normal subgroup of $G$. Let $f:G\to G/H$ be given by $f\left(g\right)=gH$. Show that (a)   $f$ is a group homomorphism, (b)   $\ker f=H$, (c)   $\mathrm{im}f=G/H$.
	Let $f:G\to H$ be a group homomorphism. Show that $G/\ker f\cong \mathrm{im}f.$

Week 8: Examples and computations

	Let $$A=\left(\begin{array}{cc}0& 1\\ -1& -1\end{array}\right)\text{and}B=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$$ Show that $A$ has order 3, that $B$ has order 4 and that $AB$ has infinite order.
	Assume that $G$ is a group such that $$\text{if}g,h\in G\text{then}{\left(gh\right)}^2=g^2{h}^2.$$ Show that $G$ is commutative.
	Decide whether the positive integers is a subgroup of the integers with operation addition.
	Decide whether the set of permutations which fix 1 is a subgroup of $S_n$.
	List all subgroups of $\mathbb{Z}/12\mathbb{Z}$.
	Let $G$ be a group, let $H$ be a subgroup and let $g\in G$. Show that $gH{g}^{-1}=\left\{gh{g}^{-1}\right|h\in H\}$ is a subgroup of $G$.
	Let $G$ be a group and let $g\in G$. Let $f:G\to G$ be given by $f\left(h\right)=gh{g}^{-1}$. Show that $f$ is an isomorphism.
	Show that ${\mathrm{SO}}_2\left(ℝ\right)$ is isomorphic to $U_1\left(ℂ\right)$.
	Show that $(ℝ,+)$ and $({ℝ}^{\times },\times )$ are not isomorphic.
	Show that $(\mathbb{Z},+)$ and $(\mathbb{Q},+)$ are not isomorphic.
	Show that $(\mathbb{Z},+)$ and $({\mathbb{Q}}_{>0},\times )$ are not isomorphic.
	Show that ${\mathrm{SL}}_2\left(\mathbb{Z}\right)$ is a subgroup of ${\mathrm{GL}}_2\left(ℝ\right)$.
	Find the orders of elements $1,-1,2$ and $i$ in the group ${ℂ}^{\times }=ℂ-\left\{0\right\}$ with operation multiplication.
	Find the orders of elements in $\mathbb{Z}/6\mathbb{Z}$.
	Find the subgroups of $\mathbb{Z}/6\mathbb{Z}$.
	Write the element (345) in $S_5$ in diagram notation, two line notation, and as a permutation matrix, and determine its order.
	Write the element (13425) in $S_5$ in diagram notation, two line notation, and as a permutation matrix, and determine its order.
	Write the element (13)(24) in $S_5$ in diagram notation, two line notation, and as a permutation matrix, and determine its order.
	Write the element (12)(345) in $S_5$ in diagram notation, two line notation, and as a permutation matrix, and determine its order.
	Let $n$ be a positive integer. Determine if the group of complex $n$th roots of unity $\{z\in ℂ|z^n=1\}$ (with operation multiplication) is a cyclic group.
	Determine if the rational numbers $\mathbb{Q}$ with operation addition is a cyclic group.
	Find the order of the element (1,2) in the group $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/8\mathbb{Z}$.
	Show that the group $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/6\mathbb{Z}$ and the group $\mathbb{Z}/12\mathbb{Z}$ are not isomorphic.
	Show that the group $\mathbb{Z}\times \mathbb{Z}$ and the group $\mathbb{Q}$ with operation addition are not isomorphic.
	Let $G$ be a group and let $a,b\in G$. Assume that $ab=ba$. (a)   Prove, by induction, that if $n\in {\mathbb{Z}}_{>0}$ then $a{b}^n=b^{n}a$, (b)   Prove, by induction, that if $n\in {\mathbb{Z}}_{>0}$ then $a^n{b}^n=b^n{a}^n$, (c)   Show that the order of $ab$ divides the least common multiple of the order of $a$ and the order of $b$. (d)   Show that if $a=\left(12\right)$ and $b=\left(13\right)$ then the order of $ab$ does not divide the least common multiple of the order of $a$ and the order of $b$.
	Show that the order of ${\mathrm{GL}}_2\left(\mathbb{Z}/2\mathbb{Z}\right)$ is 6.
	Let $p$ be a prime positive integer. Find the order of the group ${\mathrm{GL}}_2\left(\mathbb{Z}/p\mathbb{Z}\right)$.
	Let $n\in {\mathbb{Z}}_{>0}$ and let $p$ be a prime positive integer. Find the order of the group ${\mathrm{GL}}_n\left(\mathbb{Z}/p\mathbb{Z}\right)$.
	Show that the group $\mathbb{Z}\left[x\right]$ of polynomials with integer coefficients with operation addition is isomorphic to the group ${\mathbb{Q}}_{>0}$ with operation multiplication.
	Let $G$ be a group with less than 100 elements which has subgroups of orders 10 and 25. Find the order of $G$.
	Let $G$ be a group and let $H$ and $K$ be subgroups of $G$. Show that $|H\cap K|$ is a common divisor of $\left|H\right|$ and $\left|K\right|$.
	Let $G$ be a group and let $H$ and $K$ be subgroups of $G$. Assume that $\left|H\right|=7$ and $\left|K\right|=29$. Show that $H\cap K=\left\{1\right\}$.
	Let $H$ be the subgroup of $G=\mathbb{Z}/6\mathbb{Z}$ generated by 3. Compute the right cosets of $H$ in $G$ and the index $|G:H|$.
	Let $H$ be the subgroup of $G=\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$ generated by $(1,0)$. Find the order of each element in $G/H$ and identify the group $G/H$.
	Let $H$ be the subgroup of $G=\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$ generated by $(0,2)$. Find the order of each element in $G/H$ and identify the group $G/H$.
	Let $n\in {\mathbb{Z}}_{\ge 2}$ and define $f:{\mathrm{GL}}_n\left(ℂ\right)\to {\mathrm{GL}}_n\left(ℂ\right)$ by $f\left(A\right)=A^t$. Determine whether $f$ is a group homomorphism.
	Let $n\in {\mathbb{Z}}_{\ge 2}$ and define $f:{\mathrm{GL}}_n\left(ℂ\right)\to {\mathrm{GL}}_n\left(ℂ\right)$ by $f\left(A\right)={\left(A^{-1}\right)}^t$. Determine whether $f$ is a group homomorphism.
	Let $n\in {\mathbb{Z}}_{\ge 2}$ and define $f:{\mathrm{GL}}_n\left(ℂ\right)\to {\mathrm{GL}}_n\left(ℂ\right)$ by $f\left(A\right)=A^2$. Determine whether $f$ is a group homomorphism.
	Let $B$ be the subgroup of ${\mathrm{GL}}_2\left(ℝ\right)$ of upper triangular matrices and let $T$ be the subgroup of ${\mathrm{GL}}_2\left(ℝ\right)$ of diagonal matrices. Let $f:B\to T$ be given by $$f\left(\left(\begin{array}{cc}a& b\\ 0& c\end{array}\right)\right)=\left(\begin{array}{cc}a& 0\\ 0& c\end{array}\right).$$ Show that $f$ is a group homomorphism. Find $N=\ker f$ and identify the quotient $B/N$.
	Assume $G$ is a cyclic group and let $N$ be a subgroup of $G$. Show that $N$ is a normal subgroup of $G$ and that $G/N$ is a cyclic group.
	Simplify $3^{52}$ mod 53.
	Suppose that $2^{147052}=76511$ mod 147053. What can you conclude about 147053?
	Show that if $f:G\to H$ is a group homomorphism and $a_1,a_2,\dots ,a_n\in G$ then $f(a_1{a}_2\cdots a_n)=f\left(a_1\right)f\left(a_2\right)\cdots f\left(a_n\right)$.
	Describe all group homomorphisms $f:\mathbb{Z}\to \mathbb{Z}$.
	Show that ${\mathrm{SO}}_n\left(ℝ\right)$ is a normal subgroup of $O_n\left(ℝ\right)$ by finding a homomorphism $f:O_n\left(ℝ\right)\to \{\pm 1\}$ with kernel ${\mathrm{SO}}_n\left(ℝ\right)$. Identify the quotient $O_n\left(ℝ\right)/{\mathrm{SO}}_n\left(ℝ\right)$.
	Show that ${\mathrm{SU}}_n\left(ℂ\right)$ is a normal subgroup of $U_n\left(ℂ\right)$ by finding a homomorphism $f:U_n\left(ℂ\right)\to U_1\left(ℂ\right)$ with kernel ${\mathrm{SU}}_n\left(ℂ\right)$. Identify the quotient $U_n\left(ℂ\right)/{\mathrm{SU}}_n\left(ℂ\right)$.
	Let $G$ be a group and let $H$ be a subgroup of $G$. Let $f:G/H\to H\backslash G$ be given by $f\left(aH\right)=H{a}^{-1}$. Show that $f$ is a function and that $f$ is a bijection.
	Let $G=\mathbb{Z}$ and $H=2\mathbb{Z}$. Compute the cosets of $H$ in $G$ and the index $|G:H|$.
	Let $G=S_3$ and let $H$ be the subgroup generated by (123). Compute the cosets of $H$ in $G$ and the index $|G:H|$.
	Let $G=S_3$ and let $H$ be the subgroup generated by (12). Compute the cosets of $H$ in $G$ and the index $|G:H|$.
	Let $G={\mathrm{GL}}_2\left(ℝ\right)$ and let $H={\mathrm{SL}}_2\left(ℝ\right)$. Compute the cosets of $H$ in $G$ and the index $|G:H|$.
	Let $G$ be the subgroup of ${\mathrm{GL}}_2\left(ℝ\right)$ given by $$G=\left\{\left(\begin{array}{cc}x& y\\ 0& 1\end{array}\right)\right|x,y\in ℝ,x>0\}$$ Let $H$ be the subgroup of $G$ given by $$H=\left\{\left(\begin{array}{cc}z& 0\\ 0& 1\end{array}\right)\right|z\in ℝ,z>0\}.$$ Each element of $G$ can be identified with a point $(x,y)$ of ${ℝ}^2$. Use this to describe the right cosets of $H$ in $G$ geometrically. Do the same for the left cosets of $H$ in $G$.
	Consider the set $AX=B$ of linear equations where $X$ and $B$ are column vectors, $X$ is the matrix of unknowns, and $A$ the matrix of coefficients. Let $W$ be the subspace of ${ℝ}^n$ which is the set of solutions of the homogeneous equations $AX=0$. Show that the set of solutions of $AX=B$ is either empty or is a coset of $W$ in the group ${ℝ}^n$ (with operation addition).
	Let $H$ be a subgroup of index 2 in a group $G$. Show that if $a,b\in G$ and $a\notin H$ and $b\notin H$ then $ab\in H$.
	Let $G$ be a group. Let $H$ be a subgroup of $G$ such that if $a,b\in G$ and $a\notin H$ and $b\notin H$ then $ab\in H$. Show that $H$ has index 2 in $G$.
	Let $G$ be a group of order $841={\left(29\right)}^2$. Assume that $G$ is not cyclic. Show that if $g\in G$ then $g^{29}=1$.
	Show that the subgroup $\left\{\right(1),(123),(132\left)\right\}$ of $S_3$ is a normal subgroup.
	Show that the subgroup $\left\{\right(1),(12\left)\right\}$ of $S_3$ is not a normal subgroup.
	Show that ${\mathrm{SL}}_n\left(ℂ\right)$ is a normal subgroup of ${\mathrm{GL}}_n\left(ℂ\right)$.
	Let $G$ be a group. Show that $\left\{1\right\}$ and $G$ are normal subgroups of $G$.
	Show that every subgroup of an abelian group is normal.
	Write down the cosets in ${\mathrm{GL}}_n\left(ℂ\right)/{\mathrm{SL}}_n\left(ℂ\right)$ then show that $${\mathrm{GL}}_n\left(ℂ\right)/{\mathrm{SL}}_n\left(ℂ\right)\simeq {\mathrm{GL}}_1\left(ℂ\right).$$
	Show that the function $\det :{\mathrm{GL}}_n\left(ℂ\right)\to {\mathrm{GL}}_1\left(ℂ\right)$ given by taking the determinant of a matrix is a homomorphism.
	Show that the function $f:{\mathrm{GL}}_1\left(ℂ\right)\to {\mathrm{GL}}_1\left(ℝ\right)$ given by $f\left(z\right)=\left|z\right|$ is a homomorphism.
	Show that the determinant function $\det :{\mathrm{GL}}_n\left(ℂ\right)\to {\mathrm{GL}}_1\left(ℂ\right)$ is surjective and has kernel ${\mathrm{SL}}_n\left(ℂ\right)$.
	Show that the homomorphism $f:{\mathrm{GL}}_1\left(ℂ\right)\to {\mathrm{GL}}_1\left(ℝ\right)$ given by $f\left(z\right)=\left|z\right|$ has image ${ℝ}_{>0}$ and kernel $U_1\left(ℂ\right)$ (the group of $1\times 1$ unitary matrices. Conclude that $${\mathrm{GL}}_1\left(ℂ\right)/U_1\left(ℂ\right)\simeq {ℝ}_{>0}.$$
	Show that the homomorphism $$\begin{array}{cccc}f:& ℝ& \to & {\mathrm{SO}}_2\left(ℝ\right)\\ & \theta & \to & \left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right)\end{array}$$ is surjective with kernel $2\pi \mathbb{Z}$. Conclude that $$ℝ/\left(2\pi \mathbb{Z}\right)\simeq {\mathrm{SO}}_2\left(ℝ\right).$$
	Show that the set of matrices $H=\left\{\left(\begin{array}{cc}a& b\\ 0& d\end{array}\right)\right|ad\ne 0\}$ is a subgroup of GL2(ℝ) and that the set of matrices $K=\left\{\left(\begin{array}{cc}1& b\\ 0& 1\end{array}\right)\right|b\in ℝ\}$ is a normal subgroup of $H$.
	Let $G$ be a group and let $H$ be a subgroup of $G$. Show that $HH=H$.
	Let $G$ be a group and let $K$ and $L$ be normal subgroups of $G$. Show that $K\cap L$ is a normal subgroup of $G$.
	Let $G$ be a group and let $n$ be a positive integer. Assume that $H$ is the only subgroup of $G$ of order $n$. Show that $H$ is a normal subgroup of $G$.
	Let $G$ be an abelian group and let $N$ be a normal subgroup of $G$. Show that $G/N$ is abelian.
	Let $G$ be a cyclic group and let $N$ be a normal subgroup of $G$. Show that $G/N$ is cyclic.
	Find surjective homomorphisms from $\mathbb{Z}/8\mathbb{Z}$ to $\mathbb{Z}/8\mathbb{Z}$, $\mathbb{Z}/4\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}$, and $\left\{1\right\}$ (the group with one element).
	Let $ℝ$ denote the group of real numbers with the operation of addition and let $\mathbb{Q}$ and $\mathbb{Z}$ be the subgroups of rational numbers and integers, respectively. Show that it is possible to regard $\mathbb{Q}/\mathbb{Z}$ as a subgroup of $ℝ/\mathbb{Z}$ and show that this subgroup consists exactly of the elements of finite order in $ℝ/\mathbb{Z}$.

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.