\( \def\Fix{\mathrm{Fix}} \def\D{\mathsf{D}} \def\C{\mathsf{C}} \def\D{\mathsf{D}} \def\GL{\mathsf{GL}} \def\OO{\mathsf{O}} \def\SO{\mathsf{SO}} \def\RR{\mathbb{R}} \def\ZZ{\mathbb{Z}} \def\Aut{\mathrm{Aut}} \def\xx{\mathbf{x}} \)
jam - wiki

Feedback on the 2019 exam:

You can see the exam here

Question 1

(a) [6 marks] Most obtained 6/6 for this, though a few lost a mark for saying that \(\D_2\) is not Abelian.

(b) [6 marks] Many students didn't read the question! The question defined and asked for the `extended symmetry group' for the two functions. So many lost a mark for each of \(f\) and \(g\) if they failed to include the \(\sigma\) term. Some wrote that \(y\mapsto -y\) combined with \(\sigma=-1\) is a symmetry---but the question talks about transformations of \(x\in\RR\), and \(y\mapsto-y\) is not one of those! (Indeed, \(\sigma=-1\) is \(y\mapsto -y\).)

Question 2

(a) [5 marks] This was done well by almost everyone.

(b) [3 marks] Some easy marks.

(c) [6 marks] I was pleased to see how many successfully proved that \(G_{x,y}\) is a left coset of \(G_x\) --- a few proofs went a little awry, but nothing systematic worth reporting here. Moreover a good few gave an excellent brief description of how that is used to prove the orbit-stabilizer theorem.

(d) [4 marks] The answer is \(\frac{24}{12}=2\), since there are 12 diagonals in total and any one can be taken into any other by rotating the cube. A few gave answers like \(\frac{24}{6}=4\) or \(\frac{24}{8}=3\), with no explanation of where the 6 or 8 came from. This lost 3 marks (I, rather generously, gave 1 mark just for knowing ho to use the orbit-stabilizer theorem, even though there was no explanation).

Question 3

(a) [6 marks] Almost everyone knew how to derive the product of 2 Seitz symbols correctly. And most correctly found the formula for the inverse. A few ended up with \(\mathbf{v}A^{-1}\), which doesn't make sense (the vector has the wrong shape), and lost 1 or 2 marks.

(b) (i) [6 marks] This was done well, although a few missed some points on the lattice diagram and lost 1 or 2 marks depending how bad it was. Unfortunately, getting the diagram wrong often led to being unable to find suitable \(\mathbf{a},\mathbf{b}\) for part (ii).

(ii) [8 marks] Most correctly identified suitable vectors \(\mathbf{a}\) and \(\mathbf{b}\), although many didn't address the question of whether \(\mathbf{a}\) was the shortest etc. (losing up to 2 marks). As every year (and in spite of my warnings in lectures and previous feedback), some showed \(\ZZ\{\mathbf{a},\mathbf{b}\}\subset L\) twice, but not \(L\subset \ZZ\{\mathbf{a},\mathbf{b}\}\), thus losing 2 marks. Finally, two or three students put \(\mathbf{a}=0\), but that can't we right as we require \(\{\mathbf{a},\mathbf{b}\}\) to be linearly independent.
(iii) [2 marks] If (ii) was done correctly, then this was simply a case of remembering the conditions, and most got this correct (there were a couple of guesses, which got no marks).

Question 4

(a) [4 marks] \(S_3\) is generated by 2 elements (eg, a transposition and a 3-cycle, or two transpositions, but not two 3-cycles which only generates \(A_3\)), and 1 mark was lost if equivariance was only checked for one generator.

(b) [4 marks] Only a few got 4/4 on this. The main reason was that only a few bothered to answer the question and `state carefully' the conservation of symmetry theorem. A few lost another mark for not saying exactly how that theorem was being applied.

(c) [6 marks] Some only looked for equilibria on the diagonal \(x=y=z\), of which there are two (gaining just 1 mark). Whereas in all there are 5 equilibria forming 3 group orbits. The Burnside part give the final 2 marks.

(d) [4 marks] It was sufficient to show that \(\dot x >0\) throughout the interval \(0<x<1\) showing (together with the invariance of the diagonal) that the limit had to be (1,1,1), and not (0,0,0). Many solved this by solving the ode: well done! But that was obviously more work. (Some just guessed the answer, but got no marks for that.)

Question 5

(a) [3 marks] Most remembered the definition correctly --- well done! Some forgot \(g\) (for example) and wrote \(\gamma(t) = \gamma(t+\theta T)\), getting just 1 out of 3 marks.

(b) [7 marks] I was pleased a good number of students wrote a complete solution to this and got the full 7 out of 7. Several wrote unintelligible things, such as for example \(\gamma(x(t),\,y(t)) = (x(t),\,-y(t)) \) (not clear what that might mean---\(\gamma\) is a function of \(t\) not of \(x,y\)). Some couldn't distinguish between \(\C_2\) and \(\widetilde{\C_2}\) (and similarly for \(\widetilde{\D_1}\)), which led to them getting 0 for this part. A few just guessed at which experiment was \(\delta\) and which was \(\gamma\) (in fact experiment 1 is of \(\delta\)), and lost 3 marks for that.