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M A T H 0052 G R O U PS AND G E O M E T R Y F I NA L O N L I N E

I n s t r u c t i on s :

(a) Attempt all questions.

(b) Work neatly. Points may be deducted for messy work. It is strongly recommended that you type your solutions (LATEXor Word).

(c) Your answers must be carefully but succinctly explained, using complete sentences. A correct but unjustiﬁed answer may attract only half-marks. An incorrect and unjustiﬁed answer will probably be worth zero.

(d) When explaining your answers you may make reference to the results in the printed notes (but you must be precise, providing the theorem or page number).

Q u es t i on 1 . The diagram shows a cube with each face bisected by a line parallel to two of its sides:

Use the orbit-stabiliser theorem to calculate the order of the of this ﬁgure (reﬂections allowed).

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Q u es t i on 2 .

(a) Let P be a regular heptagon (seven sides). What is the symmetry group of P (including reﬂections)?

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(b) A two-colouring of P is a colouring of each edge of P either yellow or blue. Two two-colourings are called equivalent if one is obtained from the other by a symmetry of P . How many inequivalent two- colourings of P are there?

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Q u es t i on 3 .

(a) In two dimensions a glide reﬂection in the direction of the x-axis is the composite of a translation in the direction of the x-axis and a reﬂection in the x-axis. Express such a glide reﬂection as a product of reﬂections.

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(b) Is it true that every rotation of R4 has an axis? Justify your answer with a proof or a counterexample.

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Q u es t i on 4 .

(a) Let p and q be integers > 2. Let T be a triangle with angles π/2, π/p and π/q. For which values of p and q is T spherical, euclidean, or hyperbolic?

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(b) Suppose that a ﬁnite set of spherical line-segments divide the surface of S2 into N congruent equilateral triangles and that the angle α at any vertex of any triangle lies in (0, π). Classify all such tesselations of S2 by equilateral triangles, giving the values of N and α in each case.

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Q u es t i on 5 . Show that if ρ < 1, the set

| z - i |

is a hyperbolic circle

{z e H : d 扛 (z, i) = R}

(where d 扛 is hyperbolic distance in the upper half-space model of the hy- perbolic plane). Calculate R in terms of ρ .

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Q u es t i on 6 . Consider the action of the group PSL(2, C) on the Riemann sphere S2 by M¨obius transformations. Suppose that C c S2 is a circle and that g e PSL(2, C) maps C to itself, g(C) = C. Show that the trace of g must be real or imaginary.

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Q u es t i on 7 .

(a) Let R1 and R2 be reﬂections in Rn in orthogonal mirrors Π 1 and Π2 , which you may take to contain the origin. Show that R1R2 = R2R1 . Explain why this is a half-turn about the axis Π 1 n Π2 .

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(b) In R3, describe, as precisely as you can, the composite of two half- turns about distinct axes, distinguishing carefully the diﬀerent pos- sible cases. [If the two axes do not meet and are not parallel, you may ﬁnd it helpful to make use of the unique line which meets both axes at 90 О .]

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In answering this question, bear in mind that the composite of two half- turns must be an isometry. If, for example, this isometry comes out to be a rotation, when you ‘describe’ it you will need to give its axis and angle. Similar information needs to be given for other types of isometries.

One approach is to write the half-turns as products of reﬂections in ap- propriately chosen pairs of orthgonal planes—but you may use any method.

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