Group Theory Exam 2020 - N/A PDF

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UNIVERSITY OF BRISTOL School of Mathematics GROUP THEORY MATH 33300 (Paper code MATH–33300J)

January 2020 2 hours 30 minutes

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GpTh-20

1. (a) (6 marks) Define the terms i. ii. iii. iv.

left coset, normal subgroup, cyclic group, index of a subgroup.

(b) (4 marks) Calculate the number of elements of order 2 in the group S3 × Z6 . Please explain your answer. (c) (5 marks) Let G be a group. Prove that G is finite if and only if it has a finite number of subgroups. (d) (2 marks) State the Homomorphism Theorem (without proof). (e) (8 marks) Let G be a finite group with the property that there exists an integer n > 2 such that (xy )n = xn y n for all x, y ∈ G. Consider the following subsets of G: A = {g ∈ G : g n = e}, B = {g n : g ∈ G}. i. Prove that A and B are normal subgroups of G. ii. Using part 1(d), or otherwise, prove that |G : A| = |B |. 2. (a) (6 marks) Define the terms i. ii. iii. iv.

abelian group, centre of a group, conjugacy class of an element, centraliser of an element.

(b) (4 marks) Let G be a group such that the quotient group G/Z(G) is cyclic, where Z(G) is the centre of G. Prove that G is abelian. (c) (7 marks) Let G be a finite group and let k(G) be the number of distinct conjugacy classes of G. Set k(G) α(G) = . |G| i. Calculate α(Z2 × Z2 ) and α(S4 ). ii. Assume G is non-abelian. Using part 2(b), or otherwise, prove that α(G) 6 85. (d) (8 marks) Determine the abelian groups of order 108, up to isomorphism. Which of these groups contain a cyclic subgroup of order 18? Continued... Page 2 of 4

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GpTh-20

3. (a) (6 marks) Define the terms i. ii. iii. iv.

group action, faithful action, transitive action, simple group.

(b) (9 marks) Consider the action of the cyclic group G = h(1, 4)(2, 3, 5)i < S5 on the set X of subsets of {1, . . . , 5} of size 2 given by g · {i, j } = {g(i), g(j)} for all g ∈ G, {i, j } ∈ X. i. Calculate the orbit and stabiliser of {2, 3} ∈ X . ii. Give a complete set of orbit representatives for the action of G on X . iii. Is the action of G on X faithful? Please explain your answer. (c) (2 marks) State Cauchy’s Theorem (without proof). (d) (8 marks) Let G be a group and recall that G acts on X = G by left multiplication, i.e. g · x = gx for all g ∈ G, x ∈ X. Let ϕ : G → S(X) be the corresponding homomorphism. i. Prove that this action is faithful and transitive (you do not need to verify that it is an action). ii. Let G be a finite group of order 2m, where m > 3 is odd. By considering the above homomorphism ϕ, or otherwise, prove that G is not simple.

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GpTh-20

4. (a) (4 marks) Define the terms i. p-group, ii. Sylow p-subgroup. (b) (4 marks) State Sylow’s First and Third Theorems (without proof). (c) (8 marks) Let G = A5 be the alternating group of degree 5. i. For each relevant prime p, find a generating set for a Sylow p-subgroup P of G, and determine the structure of P up to isomorphism. ii. For each such p, calculate the number of Sylow p-subgroups of G. (d) (4 marks) Using Sylow’s Theorems, prove that there is no simple group of order 224 = 25 .7. (e) (5 marks) Let G be a finite group and let H 6 G be a subgroup such that |H| is divisible by a prime p. Let P be a Sylow p-subgroup of G and consider the action of H on X = G/P given by h · gP = hgP for all h ∈ H, gP ∈ X. i. Show that H ∩ gP g −1 is the stabiliser of gP ∈ X with respect to the action of H on X . ii. Prove that there exists an element g ∈ G such that H ∩ gP g −1 is a Sylow psubgroup of H .

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