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(8) Let a E G of order k. If for integers m, n we have a prove that m = n mod k. (9) Let a, b E G. Prove that o(a) = o(a-¹) and o(ab) = o(ba). (10) 'Let G be a group of even order. Prove that G has an element of order 2. ·problems donc guide ?? = an then

(8) Let a E G of order k. If for integers m, n we have a prove that m = n mod k. (9) Let a, b E G. Prove that o(a) = o(a-¹) and o(ab) = o(ba). (10) 'Let G be a group of even order. Prove that G has an element of order 2. ·problems donc guide ?? = an then

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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TUTORIAL 1: BASIC FACTS ON GROUPS Dr. A Sacidi Group structures, University of Limpopo, 2022 In Questions 1-10 below, G is a finite group and e is the identity element. By o(a), we mean the order of the element a € G. That is, the least positive integer m with a" = e. (1) Prove that if G = p is a prime, then G has no non-trivial subgroups (that is, the only subgroups of G are (e) and G), (2) If a € G, then a² = e if and only if a = a-¹. (3) If a, b E G then prove that (b-¹ab)" = b-lab for every integer n. (4) Prove that G is abelian if for all a € G, we have a² = e (Hint: Use Problem 2). (5) Show with an example that the union of two subgroups is not always a subgroup. (6) Let Z(G) = {r € Glax = xa}. Prove that Z(G) is an abelian sub- group of G (this subgroup is called the center of G). (7) Let a E G be of order m and let k be a positive integer. Prove that m/k if and only if ak = e (Hint: Use the division algorithm). E (8) Let a € G of order k. If for integers m, n we have a = a" then prove that m = n mod k. (9) Let a, b E G. Prove that o(a) = o(a-¹) and o(ab) = o(ba). (10) 'Let G be a group of even order. Prove that G has an element of order 2. problems done guide ?? 1This one is hard. 20 21 22 24 1 29 30 33 on
Transcribed Image Text:TUTORIAL 1: BASIC FACTS ON GROUPS Dr. A Sacidi Group structures, University of Limpopo, 2022 In Questions 1-10 below, G is a finite group and e is the identity element. By o(a), we mean the order of the element a € G. That is, the least positive integer m with a" = e. (1) Prove that if G = p is a prime, then G has no non-trivial subgroups (that is, the only subgroups of G are (e) and G), (2) If a € G, then a² = e if and only if a = a-¹. (3) If a, b E G then prove that (b-¹ab)" = b-lab for every integer n. (4) Prove that G is abelian if for all a € G, we have a² = e (Hint: Use Problem 2). (5) Show with an example that the union of two subgroups is not always a subgroup. (6) Let Z(G) = {r € Glax = xa}. Prove that Z(G) is an abelian sub- group of G (this subgroup is called the center of G). (7) Let a E G be of order m and let k be a positive integer. Prove that m/k if and only if ak = e (Hint: Use the division algorithm). E (8) Let a € G of order k. If for integers m, n we have a = a" then prove that m = n mod k. (9) Let a, b E G. Prove that o(a) = o(a-¹) and o(ab) = o(ba). (10) 'Let G be a group of even order. Prove that G has an element of order 2. problems done guide ?? 1This one is hard. 20 21 22 24 1 29 30 33 on
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