Final Exam - MATH 237 数学微积分代写 Question 2 (i) Cartesian coordinates (ii) Cylindrical coordinates (iii) Spherical coordinates (b) Evaluate one of the three integrals from part (a). Ques...View details
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.
Let G be a group and let X be an arbitrary subset of G. Define what is meant by the subgroup generated by X. 
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. 
(c) Is G abelian? Justify your answer. 
(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. 
(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. 
(a) List, up to isomorphism, the abelian groups of order 900. State which major theorem from the lectures asserts correctness of your list. 
(b) Consider the group
G = Z2 ⊕ Z2 ⊕ Z9 ⊕ Z25.
(i) How many elements of order 2 are there in G? Justify your answer. 
(ii) How many elements of order 100 are there in G? Justify your answer. 
(iii) Prove that every subgroup of G of order 50 is cyclic. How many such subgroups are there in G? 
(iv) Write down a Sylow 5-subgroup of G. Justify your answer. 
(c) Prove, without using the major theorem from (a), that the group G and the group Z2 ⊕ Z2 ⊕ Z3 ⊕ Z3 ⊕ Z25 are not isomorphic. 
(a) State, without proof, the Class Equation, and define each term in it. 
(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. 
(c) Use (b) to prove that every non-trivial p-group has a non-trivial centre. 
(d) What kind of correspondence does the Correspondence Theorem assert, and between which sets? 
(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).) 
(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. 
(b) Prove that there is no simple group of order 22 · 3 · 7. 
(c) Does there exist a simple group of order 22 · 3 · 11? Justify your answer. 
(d) Does there exist a simple group of order 22 · 3 · 5? Justify your answer.