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PRIFYSGOL CYMRU/UNIVERSITY OF WALES
ABERYSTWYTH
INSTITUTE OF MATHEMATICAL AND PHYSICAL SCIENCES
SEMESTER 2 EXAMINATIONS, MAY/JUNE 2008
MA30110 - Group Theory
Time allowed - 2 hours
- All questions may be attempted.
- Marks gained from questions in section B will be given greater consideration in assessing a first class performance.
- Calculators are permitted, provided they are silent, self-powered, without communi- cations facilities, and incapable of holding text or other material that could be used to give a candidate an unfair advantage. They must be made available on request for inspection by invigilators, who are authorised to remove any suspect calculators.
Section A
- Express the permutation
π =
[
]
(a) as a product of disjoint cycles, (b) as a product of transpositions.
What is the order of π? [8 marks]
- For the permutations
ρ =
[
]
, σ =
[
]
find a permutation π so that πρπ−^1 = σ. [8 marks]
- Which of the following subsets of the symmetric group S(6), acting as usual on { 1 , 2 , 3 , 4 , 5 , 6 }, are subgroups? If so, which of them are normal? Justify your answer.
(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]
In the symmetric group S(3) let H be the subgroup generated by the transposition (12). How many distinct left cosets of H are there? Compute them explicitly. [8 marks]
State Lagrange’s theorem and define the index of a subgroup.
Suppose that a group G of order 250 has an element a of order 25 and an element b of order 10. Explain why the subgroup 〈a, b〉 has index at most 5 in G. [8 marks]
- Let H be a subgroup of a finite group G. The normaliser NG(H) is defined as
NG(H) = {g ∈ G : gH = Hg}
Show that
Section B
- Prove that a subgroup of a cyclic group is cyclic.
Let 〈a〉 be a cyclic group of order 50. Express 〈a^15 , a^20 〉 in the form 〈an〉. [7 marks]
Prove that 〈a, b : a^3 = e, b^2 = e, ab = ba〉 is a presentation of the cyclic group C 6. [10 marks]
Show that the group with presentation
〈a, b : b^2 = e, ba = a−^1 b〉
is infinite. Hint: Consider the matrices [ 1 1 0 1
]
[
]
[10 marks]
- State the Homomorphism Theorem.
For the cyclic groups C 20 = 〈a〉 and C 5 = 〈b〉, show that there exists a unique homomorphism φ : C 20 → C 5 with the property φ(a) = b. Determine ker φ and im φ explicitly and write down what the Homomorphism Theorem tells us in this case. [8 marks]
Prove that if H and K are proper subgroups of a finite group G then there exists an element in G which is not in H or K. [8 marks]
State the Orbit Stabiliser Theorem.
The symmetric group S(4) acts on R^4 by permuting the coordinates. De- scribe the stabiliser and the orbit of (2, 2 , 3 , 3) ∈ R^4 and verify the Orbit Stabiliser Theorem in this example. [7 marks]
- Let p, q be primes with p > q. Use the Sylow theorems to show that a group of order pq has a normal subgroup of order p. [10 marks]
