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This is the Exam of Group Theory and its key important points are: Order, Permutation, Disjoint Cycles, Product of Transpositions, Conjugate, Disjoint Cycle Decompositions, Even Permutations, Odd Permutations, Cyclic Permutations, Distinct Left Cosets
Typology: Exams
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MA30110 - Group Theory
Time allowed - 2 hours
(1) (a) Express the permutation
π =
as a product of disjoint cycles, as a product of transpositions. (b) What is the order and the sign of π? (c) Give an argument why all permutations of order 15 in the symmetric group S(8) are even. (d) How can you find out from disjoint cycle decompositions whether two permu- tations are conjugate to each other or not? (e) Show that
ρ =
is conjugate to π (from (a)) and find a permutation α such that απα−^1 = ρ. (20 marks)
(2) Which of the following subsets of the symmetric group S(8), acting as usual on { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 }, are subgroups? Give a short justification for your answer in each case.
(a) the set of even permutations (b) the set of odd permutations (c) the set of all permutations which fix 3 (d) the set of all cyclic permutations
(10 marks)
(3) (a) State the definition for a left coset of a subgroup H in a group G.
(b) In G = C 12 = 〈x〉, a cyclic group with 12 elements generated by x, consider the subgroup H = 〈x^3 〉 generated by x^3. How many distinct left cosets of H are there? Compute them explicitly. (10 marks)
(4) (a) State the definition of a group action and the definitions of orbits and stabilis- ers. State the Orbit-Stabiliser Theorem and Burnside’s Lemma. We colour the vertices of a square with three different colours: red (r), blue (b), green (g) and we do not distinguish colourings if they can be obtained from each other by rotations or reflections.
(8) (a) Describe the conjugation action of a group on itself and show that for this action and for any element the stabiliser is equal to the centraliser (= the subgroup of all elements commuting with the given element). (b) In the symmetric group S(4) find all elements in the orbit of the 3-cycle (123) under the conjugation action. (c) Use the Orbit-Stabiliser theorem to compute the order of the centraliser of the 3-cycle (123) and then find all the elements of this centraliser explicitly. (10 marks)
(9) (a) State the definition of a Sylow-p-subgroup. What do the Sylow theorems tell us about the numbers of Sylow-p-subgroups? (b) For a group of order 6, work out all possibilities for the numbers of Sylow-p- subgroups. (c) By giving examples of suitable groups of order 6, show that all the possibilities worked out in (b) can actually be realised. (10 marks)
