(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.

(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 ${ℂ}^n$ by $(X,Y)=XA{Y}^{*}$, where $X,Y\in {ℂ}^n$ 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.

(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.

(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 $S_3$.

(a)   Describe the full group $G$ of symmetries of this pattern.

(b)   Describe the stabiliser $H$ of one of the symbols $Y$.

(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 $H$.

(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).

(a)   What are the possible orders of subgroups of $G$?

(b)   What are the possible orders of non-cyclic subgroups of $G$?

(a)   Show that the set $$\left\{\left(\begin{array}{cc}a& b\\ 0& 1\end{array}\right)\right|a,b\in ℝ,a\ne 0\}$$ forms a group $G$ under matrix multiplication.

(b)   Show that the function $f:G\to {ℝ}^{*}$ 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 ${ℝ}^{*}$ of non-zero real numbers.

(c)   Find the image and kernel of $f$.

(a)   What are the possible sizes of orbits?

(b)   Show that there must be a point of $X$ fixed by all elements of $G$.

(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 $T$.

(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.

(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 $G$.

{   (1), (12)(34), (13)(24), (14)(23)
  (123), (132), (124), (142), (134), (143), (234), (243)   }

(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?

(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 $G/Z$?

(a)   Find the order of each element in $G$.

(b)   Is the group cyclic?

(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 $S_3$.

(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 $H\cap K$?

(a)   Show that $$G=\left\{\left(\begin{array}{ccc}1& a& c\\ 0& 1& b\\ 0& 0& 1\end{array}\right)\right|a,b,c\in ℝ\}$$ is a subgroup of ${\mathrm{GL}}_3\left(ℝ\right)$ using matrix multiplication as the operation.

(b)   Find the centre of $G$.

(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.

(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 $f$.

(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 $B^2=A$.

(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.

(a)   Show that if $f$ and $g$ are isometries of ${𝔼}^2$ then $$\mathrm{Fix}\left(gf{g}^{-1}\right)=g\mathrm{Fix}\left(f\right).$$

(b)   The non-identity isometries of ${𝔼}^2$ 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 $g\left(p\right)$.

(a)   Show that every element of the quotient group $\mathbb{Q}/\mathbb{Z}$ has finite order.

(b)   Let $S^1=\{z\in ℂ|\left|z\right|=1\}$ denote the multiplicative group of complex numbers of absolute value one. Show that the function $f:\mathbb{Q}\to S^1$ defined by $$f\left(x\right)=e^{2\pi ix}=\cos \left(2\pi x\right)+i\sin \left(2\pi x\right)$$ 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 $S^1$?

(a)   Use the Euclidean algorithm to find $d=\gcd (469,959)$.

(b)   Find integers $x,y$ such that $469x+959y=d$.

(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?

(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 $S_4$.

(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.

(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 $G$.

(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.

(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 $g^2\ne g$.

(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 $G/H$.

(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 $H$.

(a)   Consider the 1-dimensional subspaces of $V$. Show that there are exactly $p+1$ such subspaces: spanned by the vectors $$\left(\begin{array}{c}0\\ 1\end{array}\right),\left(\begin{array}{c}1\\ 1\end{array}\right),\dots \left(\begin{array}{c}p-1\\ 1\end{array}\right)\text{and}\left(\begin{array}{c}1\\ 0\end{array}\right).$$

(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 $$K=\left\{\left(\begin{array}{cc}a& 0\\ 0& a\end{array}\right)\right|a\in \mathbb{Z}/p\mathbb{Z}-\left\{0\right\}\}.$$ 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 $S_4$.