answer #5 handwritten 1. Let H and K be a subgroups of a group G. Define the relationas follows:~on Gab if and only if a = hbk, for some h H and ke KProve that is an equivalence relation on G.2. Let R+ denotes the set of positive real numbers and let f: R+R+be the bijection defined by f(x) = 3x, for x > 0. Let denote theordinary real number multiplication and let be the binary operationon R+ such that f: (R+,.) → (R+, ) is a group isomorphism..(a) If x, y E R+, find a formula for ry. What is the identity elementof (R+,)?(b) For x € R+, find a formula for the inverse of a under >.3. If a group G has finitely many subgroups, does this mean that G isfinite group? Justify your answer.4. Let p and q be distinct prime numbers and let n = p²q². Determineall the subgroups of Zn and draw the lattice diagram for subgroups.Justify your work.5. Let G be a cyclic group of order n, generated by a. Let m be an integersuch that ged(m, n) = 1. Prove that there exists a unique 2 € G suchthat x = a.6. Let n 22 be an integer.(a) If HSn and H has an odd order, then H≤ An.(b) If a, 3 € Sn, prove that either both aßa¹ and 3 are even or bothare odd.

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Answered: 5. Let G be a cyclic group of order n,…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.3: Subgroups

Problem 5E: 5. Exercise of section shows that is a group under multiplication. a. List the elements of the...

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Elements Of Modern Algebra

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5. Let G be a cyclic group of order n, generated by a. Let m be an integer such that ged(m, n) = 1. Prove that there exists a unique r € G such that x = a.