Consider the permutation group $G=\left\{\right(1\left),\right(12\left)\right(34\left),\right(13\left)\right(24\left),\right(14\left)\right(23\left)\right\}$
acting on a set $X$ of four symbols 1,2,3,4.
(a)
Describe the orbit and stabiliser of 1. Explain how the orbit/stabiliser theorem
connects $G$ and the orit and stabiliser.
(b)
Find the orbit and stabiliser of 1 for the action of the subgroup
$H=\left\{\right(1\left),\right(12\left)\right(34\left)\right\}$
acting on the set $X$.
(a)
If a group of order 9 acts on a set $X$ with 4 elements, explain why each
orbit must consist of either one or three points.
(b)
Explain why a group with 9 elements must have an element in the centre, which is different
from the identity element.
Let $V$ be a complex finite dimensional inner product space and let
$f:V\to V$ be a linear transformation
satisfying ${f}^{*}f=f{f}^{*}$.
(a)
State the spectral theorem and deduce that there is an orthonormal basis of $V$
consisting of eigenvectors of $f$.
(b)
Show that there is a linear transformation $g:V\to V$
so that $f={g}^{2}$.
(c)
Show that if every eigenvalue of $f$ has absolute value 1, then
${f}^{*}={f}^{-1}$.
(d)
Give an example to show that the result in (a) can fail if $V$ is a real inner product space.
(a)
Let $A$ be an $n\times n$ complex Hermitian matrix.
Define a product on ${\u2102}^{n}$ by
$(X,Y)=XA{Y}^{*}$, where $X,Y\in {\u2102}^{n}$ are written as row vectors. Show that this
is an inner product if all the eigenvalues of $A$ are positive real numbers.
(b)
Show that if $A={B}^{*}B$, where
$B$ is any invertible $n\times n$ complex
matrix, then $A$ is a Hermitian matrix and all the eigenvalues of $A$
are real and positive.
Let $G$ be the multiplicative group ${\mathrm{GL}}_{2}\left({\mathbb{F}}_{2}\right)$ of invertible
$2\times 2$ matrices, where the entries are from
the field with two elements
${\mathbb{F}}_{2}=\mathbb{Z}/2\mathbb{Z}$.
There are six matrices which are elements of this group.
Let $V$ be the 2-dimensional vector space over the field
${\mathbb{F}}_{2}$ ($V$ contains 4 vectors).
Then $G$ acts on $V$ by usual multiplication of column vectors by
matrices;
$A:X\to AX$, where
$A\in G$, $X\in V$.
(a)
Find the orbits and stabilisers of the vectors
${(0,0)}^{t}$
and
${(1,0)}^{t}$
under the action of $G$, where the transpose $t$ converts
row vectors to column vectors.
(b)
Use this action to construct a homomorphism $\phi $ from $G$
into ${S}_{4}$, the permutation group on 4 symbols.
(c)
Prove that the homomorphism $\phi $ is injective.
Consider the symmetric group ${S}_{4}$ acting on the four
numbers {1,2,3,4}. Consider the three ways of dividing these numbers into two pairs, namely
${P}_{1}=\left\{\right\{1,2\},\{3,4\left\}\right\}$,
${P}_{2}=\left\{\right\{1,3\},\{2,4\left\}\right\}$,
${P}_{3}=\left\{\right\{1,4\},\{2,3\left\}\right\}$.
(a)
Construct a homomorphism from ${S}_{4}$ onto
${S}_{3}$ by using the action of ${S}_{4}$
on {1,2,3,4} to give an action of ${S}_{4}$ on the set of
three objects $\{{P}_{1},{P}_{2},{P}_{3}\}$.
In particular, explain why the mapping you have described is a homomorphism.
(b)
Describe the elements of the kernel $K$ of this homomorphism and explain why this
subgroup is normal.
(c)
Explain why the quotient group ${S}_{4}/K$ is
isomorphic to ${S}_{3}$.
Consider the infinte pattern of symbols
$$\cdots {\scriptscriptstyle YYYYYYYYYYY}\cdots $$
(a)
Describe the full group $G$ of symmetries of this pattern.
(b)
Describe the stabiliser $H$ of one of the symbols
${\scriptscriptstyle Y}$.
(c)
Describe the maximal normal subgroup of translations $T$ in $G$
and explain why the quotient group $G/T$ is isomorphic to the
stabiliser subgroup $H$.
An inner product $\u27e8,\u27e9$ on
${\mathbb{R}}^{3}$ is defined by
$$\u27e8({x}_{1},{x}_{2},{x}_{3}),({y}_{1},{y}_{2},{y}_{3})\u27e9={x}_{1}{y}_{1}+2{x}_{2}{y}_{2}+3{x}_{3}{y}_{3}.$$
Let $W$ be the subspace of ${\mathbb{R}}^{3}$
spanned by $\left\{\right(1,-1,0),(0,1,-1\left)\right\}$.
Find all vectors in $W$ orthogonal to $(1,1,-1)$.
The subset {1, 2, 4, 5, 7, 8} of $\mathbb{Z}/9\mathbb{Z}$
forms a group $G$ under multiplication modulo 9.
(a)
Show that the group $G$ is cyclic.
(b)
Give an example of a non-cyclic group of order 6.
(a)
Express the following permutations as products of disjoint cycles:
(134)(25)$\cdot $(12345) and
the inverse of (12)(3456).
(b)
Find the order of the permutation (123)(4567).
Let $G$ be a group of order 21.
(a)
What are the possible orders of subgroups of $G$?
(b)
What are the possible orders of non-cyclic subgroups of $G$?
Always explain your answers.
(a)
Show that the set
$$\left\{\left(\begin{array}{cc}a& b\\ 0& 1\end{array}\right)\phantom{\rule{0.3em}{0ex}}\right|\phantom{\rule{0.3em}{0ex}}a,b\in \mathbb{R},a\ne 0\}$$
forms a group $G$ under matrix multiplication.
(b)
Show that the function $f:G\to {\mathbb{R}}^{*}$ defined by
$$f\left(\left(\begin{array}{cc}a& b\\ 0& 1\end{array}\right)\right)={a}^{2}$$
is a homomorphism from $G$ to the multiplicative group
${\mathbb{R}}^{*}$ of non-zero real numbers.
(c)
Find the image and kernel of $f$.
A group $G$ of order 8 acts on a set $X$ consisting
of 11 points.
(a)
What are the possible sizes of orbits?
(b)
Show that there must be a point of $X$ fixed by all elements of $G$.
Always explain your answers.
Let $V$ be a complex inner product space and let $T:V\to V$ be a linear transformation such that
${T}^{*}T=T{T}^{*}$.
(a)
Explain how the adjoint ${T}^{*}$ of $T$
is defined.
(b)
Prove that $\Vert Tx\Vert =\Vert {T}^{*}x\Vert $
for all $x\in V$.
(c)
Deduce that the nullspace of ${T}^{*}$ is equal to the nullspace
of $T$.
Let $A$ be an $n\times n$ complex matrix.
(a)
Prove that if $A$ is Hermitian, then all eigenvalues of $A$
are real.
(b)
Carefully state the spectral theorem for normal matrices. Use
this to show that if $A$ is a normal matrix with all real
eigenvalues, then $A$ is Hermitian.
Let $X$ be the graph of $y=\mathrm{sin}x$
in the $x$-$y$ plane,
$$X=\left\{\right(x,y)\in {\mathbb{R}}^{3}\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}y=\mathrm{sin}x\},$$
and let $G$ be the symmetry group of $X$.
(a)
Describe all the symmetries in $G$.
(b)
Find the orbit and stabilizer of the point $(0,0)$
under the action of $G$ on $X$.
(c)
Find the translational subgroup $T$ of $G$.
(d)
Explain why $T$ is a normal subgroup of $G$.
Let $G$ be the subgroup of the symmetric group
${S}_{4}$ consisting of the permutations
(a)
Show that $G$ has 4 conjugacy classes, containing 1, 3, 4 and 4 elements.
(b)
Explain why any normal subgroup of $G$ is a union of conjugacy classes.
(c)
Deduce that $G$ contains no normal subgroup of order 6.
(d)
Does $G$ contain any subgroup of order 6?
Always explain your answers.
(a)
Show that if $G$ is a group with centre $Z$
such that $G/Z$ is cyclic, then $G$
is abelian.
(b)
If $G$ is a nonabelian group of order ${p}^{3}$
where $p$ is prime, what can you say about the centre $Z$
of $G$ and the quotient group $G/Z$?
Always explain your answers.
Let $V={\mathcal{P}}_{2}\left(\mathbb{R}\right)$
be the real vector space of all polynomials of degree $\le 2$ with
real coefficients. An inner product $\u27e8,\u27e9$
on $V$ is defined by
$$\u27e8p,q\u27e9={\int}_{-1}^{1}p\left(x\right)q\left(x\right)\phantom{\rule{0.2em}{0ex}}dx.$$
Find a basis for the orthogonal complement of the subspace $W$
spanned by $\{1,x\}$.
Consider the complex matrix
$$A=\left(\begin{array}{cc}1& 1\\ i& 1\end{array}\right).$$
Decide whether the matrix is: (i) Hermitian, (ii) unitary, (iii) normal, (iv) diagonalizable.
Always explain your answers.
The set of eight elements $\{\pm 1,\pm 2,\pm 4,\pm 7\}$
forms a group $G$ under multiplication modulo 15.
(a)
Find the order of each element in $G$.
(b)
Is the group cyclic?
Always explain your answers.
(a)
Express the permutation (1342)$\cdot $(345)(12) as a product of
disjoint cycles.
(b)
Find the order of the permutation (12)(4536) in the group ${S}_{6}$.
(c)
Find all the conjugates of (123) in the group ${S}_{3}$.
Let $G$ be a finite group containing a subgroup $H$ of
order 4 and a subgroup $K$ of order 7.
(a)
State Lagrange's theorem for finite groups.
(b)
What can you say about the order of $G?$
(c)
What can you say about the order of the subgroup $H\cap K$?
Always explain your answers.
(a)
Show that
$$G=\left\{\left(\begin{array}{ccc}1& a& c\\ 0& 1& b\\ 0& 0& 1\end{array}\right)\phantom{\rule{0.3em}{0ex}}\right|\phantom{\rule{0.3em}{0ex}}a,b,c\in \mathbb{R}\}$$
is a subgroup of ${\mathrm{GL}}_{3}\left(\mathbb{R}\right)$
using matrix multiplication as the operation.
(b)
Find the centre of $G$.
Let $G$ be the group of symmetries of the rectangle $X$
with vertices
$(2,1),(2,-1),(-2,1),(-2,-1)$.
(a)
Give geometric descriptions of the symmetries in $G$.
(b)
Find the orbit and stabilizer of the point $Q=(2,0)$ under the action of
$G$ on $X$.
(c)
Check that your answers to parts (a) and (b) are consistent with the orbit-stabiliser theorem.
Let $f:V\to V$ be a linear operator
on a finite dimensional inner product space.
(a)
Explain how the adjoint ${f}^{*}$ of $f$
is defined.
(b)
Prove that the nullspace of ${f}^{*}$ is the orthogonal complement
of the range of $f$.
(c)
Deduce that the nullity of ${f}^{*}$ is equal to the
nullity of $f$.
Consider the complex matrix
$$A=\left(\begin{array}{cc}4& -5i\\ 5i& 4\end{array}\right).$$
(a)
Without calculating eigenvalues, explain why $A$ is diagonalizable.
(b)
Find a diagonal matrix $D$ and a unitary matrix $U$
such that
$${U}^{-1}AU=D.$$
(c)
Write down ${U}^{-1}$.
(d)
Find a complex matrix $B$ such that ${B}^{2}=A$.
Let $G$ be a group in which every element has order 1 or 2.
(a)
Prove that $G$ is abelian.
(b)
Prove that if $G$ is finite then $G$ has order
${2}^{n}$ for some integer $n\in {\mathbb{Z}}_{\ge 0}$.
(c)
For each integer $n\ge 1$, give an example of a group of
order ${2}^{n}$ with each element of order 1 or 2.
Always explain your answers.
For any isometry $f:{\mathbb{E}}^{2}\to {\mathbb{E}}^{2}$
of the Euclidean plane, let
$$\mathrm{Fix}\left(f\right)=\{x\in {\mathbb{E}}^{2}\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}f\left(x\right)=x\}$$
denote the fixed point set of $f$.
(a)
Show that if $f$ and $g$ are isometries of
${\mathbb{E}}^{2}$ then
$$\mathrm{Fix}\left(gf{g}^{-1}\right)=g\mathrm{Fix}\left(f\right).$$
(b)
The non-identity isometries of ${\mathbb{E}}^{2}$ are of four types:
rotations, reflections, translations, and glide reflections. Describe the fixed point set for each type.
(c)
Deduce from parts (a) and (b) that if $f$ is a rotation about a point $p$
then $gf{g}^{-1}$
is a rotation about the point $g\left(p\right)$.
Let $\mathbb{Q}$ denote the additive group of rational numbers, and $\mathbb{Z}$
the subgroup of integers.
(a)
Show that every element of the quotient group $\mathbb{Q}/\mathbb{Z}$
has finite order.
(b)
Let ${S}^{1}=\{z\in \u2102\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}\left|z\right|=1\}$
denote the multiplicative group of complex numbers of absolute value one. Show that the
function $f:\mathbb{Q}\to {S}^{1}$
defined by
$$f\left(x\right)={e}^{2\pi ix}=\mathrm{cos}\left(2\pi x\right)+i\mathrm{sin}\left(2\pi x\right)$$
is a homomorphism.
(c)
Find the kernel of $f$.
(d)
Deduce that $\mathbb{Q}/\mathbb{Z}$ is isomorphic
to a subgroup of ${S}^{1}$.
(e)
Is $\mathbb{Q}/\mathbb{Z}$ is isomorphic
to ${S}^{1}$?
Always explain your answers.
(a)
Use the Euclidean algorithm to find $d=\mathrm{gcd}(469,959)$.
(b)
Find integers $x,y$ such that
$469x+959y=d$.
The complex vector space ${\u2102}^{4}$ has an inner product
defined by
$$\u27e8a,b\u27e9={a}_{1}{\stackrel{\u203e}{b}}_{1}+{a}_{2}{\stackrel{\u203e}{b}}_{2}+{a}_{3}{\stackrel{\u203e}{b}}_{3}+{a}_{4}{\stackrel{\u203e}{b}}_{4}$$
for $a=({a}_{1},{a}_{2},{a}_{3},{a}_{4})$, $b=({b}_{1},{b}_{2},{b}_{3},{b}_{4})\in {\u2102}^{4}$.
Let $W$ be the subspace of ${\u2102}^{4}$
spanned by the vectors
$(1,0,-1,0)$
and
$(0,1,0,i)$.
Find a basis for the orthogonal complement ${W}^{\perp}$ of
$W$.
Determine whether the matrix
$A=\left(\begin{array}{cc}3& 4i\\ 4i& 3\end{array}\right)$
is (i) Hermitian, (ii) unitary, (iii) normal, (iv) diagonalizable.
Always explain your answers.
The sets
${G}_{1}=\{1,3,9,11\}$
and
${G}_{2}=\{1,7,9,15\}$
form groups under multiplication modulo 16.
(a)
Find the order of each element in ${G}_{1}$ and each element
in ${G}_{2}$.
(b)
Are the groups ${G}_{1}$ and
${G}_{2}$ isomorphic?
Always explain your answers.
(a)
Express the following permutation as a product of disjoint cycles:
(234)(56)*(1354)(26).
(b)
Find the order of the permutation (12)(34567) in ${S}_{7}$.
(c)
Find all conjugates of (13)(24) in the group ${S}_{4}$.
Let $G$ be a group of order 35.
(a)
What does Lagrange's theorem tell you about the orders of subgroups of $G$?
(b)
If $H$ is a subgroup of $G$ with $H\ne G$,
expalin why $H$ is cyclic.
Consider the set of matrices
$$G=\left\{\left(\begin{array}{cc}a& b\\ b& a\end{array}\right)\phantom{\rule{.3em}{0ex}}\right|\phantom{\rule{.3em}{0ex}}a,b\in \mathbb{R},{a}^{2}-{b}^{2}=1\}.$$
Prove that $G$ is a group using matrix multiplication as the operation.
Let $X$ be a subset of ${\mathbb{R}}^{2}$
consisting of the four edges of a square together with its two diagonals. Let $Y$
be obtained from $X$ by filling in two triangles as shown below:
Let $G$ be the symmetry group of $X$ and $H$
the symmetry group of $Y$.
(a)
Describe the group $G$ by giving geometric descriptions of the symmetries in
$G$, and writing down a familiar group isomorphic to $G$.
(b)
Give a similar description of $H$.
(c)
Explain why $H$ is a normal subgroup of $G$.
Let $f:V\to V$ be a self-adjoint linear operator
on an inner product space $V$,
i.e. ${f}^{*}=f$.
(a)
Prove that every eigenvalue of $f$ is real.
(b)
Let ${v}_{1},{v}_{2}$ be
eigenvectors of $f$ corresponding to eigenvalues
${\lambda}_{1},{\lambda}_{2}$
with ${\lambda}_{1}\ne {\lambda}_{2}$.
Prove that ${v}_{1}$ and ${v}_{2}$
are orthogonal.
Let $A$ be a $6\times 6$ complex matrix
with minimal polynomial
$$m\left(X\right)={(X+1)}^{2}(X-1).$$
(a)
Describe the possible characteristic polynomials for $A$.
(b)
Let the possible Jordan normal forms for $A$ (up to reordering the Jordan
blocks).
(c)
Explain why $A$ is invertible and write
${A}^{-1}$ as a
polynomial in $A$.
(a)
Let $f:V\to V$ be a normal linear
operator on a complex inner product space $V$ such that ${f}^{4}={f}^{3}$.
Use the spectral theorem to prove that $f$ is self-adjoint and that
${f}^{2}=f$.
(b)
Give an example of a linear operator $g:V\to V$
on a complex inner product space $V$ such that ${g}^{4}={g}^{3}$ but
${g}^{2}\ne g$.
Consider the subgroup $H=\{\pm 1,\pm i\}$ of the multiplicative group $G={\u2102}^{*}$ of non-zero complex numbers.
(a)
Describe the cosets of $H$ in $G$.
Draw a diagram in the complex plane showing a typical coset.
(b)
Show that the function $f:G\to G$
defined by $f\left(z\right)={z}^{4}$
is a homomorphism and find its kernel and image.
(c)
Explain why $H$ is a normal subgroup of $G$ and identify
the quotient group $G/H$.
Let $G$ be the cyclic subgroup of ${S}_{7}$
generated by the permutation (12)(3456). Consider the action of $G$ on
$X=\{1,2,3,4,5,6,7\}$.
(a)
Write down all the elements of $G$.
(b)
Find the orbit and stabilizer of (i) 1, (ii) 3 and (iii) 7.
Check that your answers are consistent with the orbit-stabilizer theorem.
(c)
Prove that if a group $H$ of order 4 acts on a set $Y$
with 7 elements then there must be at least one element of $Y$ fixed by all
elements of $H$.
Let $p$ be a prime number, and let $V$ be the vector space
over the field $\mathbb{Z}/p\mathbb{Z}$ consisting of all
column vectors in
${\left(\mathbb{Z}/p\mathbb{Z}\right)}^{2}$:
$$V=\left\{\left(\begin{array}{c}x\\ y\end{array}\right)\phantom{\rule{.3em}{0ex}}\right|\phantom{\rule{.3em}{0ex}}x,y\in \mathbb{Z}/p\mathbb{Z}\}.$$
Let $G={\mathrm{GL}}_{2}\left(\mathbb{Z}/p\mathbb{Z}\right)$ be the group of invertible $2\times 2$
matrices with
$\mathbb{Z}/p\mathbb{Z}$
entries using matrix multiplication.This acts on $V$ by matrix
multiplication as usual: $A\cdot v=Av$ for all $A\in G$ and
all $v\in V$.
(a)
Consider the 1-dimensional subspaces of $V$. Show that there are exactly
$p+1$ such subspaces: spanned by the vectors
$$\left(\begin{array}{c}0\\ 1\end{array}\right),\phantom{\rule{.5em}{0ex}}\left(\begin{array}{c}1\\ 1\end{array}\right),\phantom{\rule{.5em}{0ex}}\dots \phantom{\rule{.5em}{0ex}}\left(\begin{array}{c}p-1\\ 1\end{array}\right)\phantom{\rule{.5em}{0ex}}\text{and}\phantom{\rule{.5em}{0ex}}\left(\begin{array}{c}1\\ 0\end{array}\right).$$
(b)
Explain why $G$ also acts on the set $X$ of
1-dimensional subspaces of $V$. This gives a homomorphism
$\phi :G\to {S}_{p+1}$.
(c)
Show that the kernel of $\phi $ consists of the scalar matrices
$$K=\left\{\left(\begin{array}{cc}a& 0\\ 0& a\end{array}\right)\phantom{\rule{.3em}{0ex}}\right|\phantom{\rule{.3em}{0ex}}a\in \mathbb{Z}/p\mathbb{Z}-\left\{0\right\}\}.$$
Deduce that the quotient group $G/K$ is isomorphic to a subgroup
of ${S}_{p+1}$.
(d)
For the case where $p=3$, find
$\left|K\right|$
and
$\left|G\right|$. Deduce that
$G/K$ is isomorphic to ${S}_{4}$.