数学群代考 MT4003代写 Groups代写 数学考试代考

MT4003 Groups

数学群代考 EXAM DURATION: 2 hours EXAM INSTRUCTIONS: Attempt ALL questions. The number in square brackets shows the maximum marks obtainable

EXAM DURATION: 2 hours

EXAM INSTRUCTIONS: Attempt ALL questions.

The number in square brackets shows the maximum marks obtainable for that question or part-question.

Your answers should contain the full working required to justify your solutions.

1. 数学群代考

(a)

Let G be a group and let X be an arbitrary subset of G. Define what is meant by the subgroup generated by X. [2]

Let G be the subgroup of the symmetric group S₆ generated by

α = (1 3)(2 4)(5 6) and β = (1 5)(2 6).

(b) Write down the orders of the elements α, β and αβ. [2]

(c) Is G abelian? Justify your answer. [1]

(d) Prove that every element of G may be expressed in the form (αβ)^k or (αβ)^kα where 0 ≤ k ≤ 5. Hence, or otherwise, prove that the order of G is 12. [4]

(e) We know from lectures that every group of order 12 is isomorphic to precisely one of:

Z₁₂, Z₂ ⊕ Z₂ ⊕ Z₃, D₆, A₄, or T.

Here D₆ is the dihedral group, A₄ is the alternating group, and T is the subgroup of S₁₂ generated by

(1 2 3 4 5 6)(7 8 9 10 11 12) and (1 7 4 10)(2 12 5 9)(3 11 6 8).

To which of these groups is G isomorphic? Justify your answer. [3]

2. 数学群代考

(a) List, up to isomorphism, the abelian groups of order 900. State which major theorem from the lectures asserts correctness of your list. [4]

(b) Consider the group

G = Z₂ ⊕ Z₂ ⊕ Z₉ ⊕ Z₂₅.

(i) How many elements of order 2 are there in G? Justify your answer. [1]

(ii) How many elements of order 100 are there in G? Justify your answer. [2]

(iii) Prove that every subgroup of G of order 50 is cyclic. How many such subgroups are there in G? [4]

(iv) Write down a Sylow 5-subgroup of G. Justify your answer. [2]

(c) Prove, without using the major theorem from (a), that the group G and the group Z₂ ⊕ Z₂ ⊕ Z₃ ⊕ Z₃ ⊕ Z₂₅ are not isomorphic. [3]

3. 数学群代考

(a) State, without proof, the Class Equation, and define each term in it. [2]

(b) Let p be a prime, and let G be a non-trivial p-group, i.e. a group of order pⁿ for some n ≥ 1. Explain why, for a proper subgroup H, both the order |H| of H and the index [G : H] of H in G are divisible by p. [2]

(c) Use (b) to prove that every non-trivial p-group has a non-trivial centre. [3]

(d) What kind of correspondence does the Correspondence Theorem assert, and between which sets? [2]

(e) Prove that a p-group G of order pⁿ with n ≥ 1 contains a normal subgroup of order pⁿ⁻¹ . (Hint: Use induction on n, and consider the centre Z(G).) [4]

4.

(a) State, without proof, the 3rd Sylow Theorem, concerning the number of Sylow p-subgroups of a finite group G. Does this theorem uniquely determine the number of Sylow p-subgroups of G? Justify your answer. [3]

(b) Prove that there is no simple group of order 2² · 3 · 7. [2]

(c) Does there exist a simple group of order 2² · 3 · 11? Justify your answer. [3]

(d) Does there exist a simple group of order 2² · 3 · 5? Justify your answer. [1]

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