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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.

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.

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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1. Let H and K be a subgroups of a group G. Define the relation as follows: ~on G ab if and only if a = hbk, for some h H and ke K Prove 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 the ordinary real number multiplication and let be the binary operation on 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 element of (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 is finite group? Justify your answer. 4. Let p and q be distinct prime numbers and let n = p²q². Determine all 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 integer such that ged(m, n) = 1. Prove that there exists a unique 2 € G such that 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 both are odd.
Transcribed Image Text:1. Let H and K be a subgroups of a group G. Define the relation as follows: ~on G ab if and only if a = hbk, for some h H and ke K Prove 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 the ordinary real number multiplication and let be the binary operation on 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 element of (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 is finite group? Justify your answer. 4. Let p and q be distinct prime numbers and let n = p²q². Determine all 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 integer such that ged(m, n) = 1. Prove that there exists a unique 2 € G such that 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 both are odd.
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