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4.14 Give an example of an infinite group G such that every element of G has finite order. 4.15 a) Find (123,321), and find integers x and y such that 123x+321y=(123,321). b) Find (862,347), and find integers x and y such that 862x+347y-(862,347). c) Find (7469,2464), and find integers x y such that 7469x+2464y= (7469,2464). 4.16 Prove that if G=(x), then G=(x¹). ted 4.17 Prove that if G=(x) and Gis inite, then x and x-are the only generators of G. 4.18 Prove parts (ii) and (iii) Theorem 4.1. 4.19 Prove part (i) of Theorem 4.4. 4.20 Let G be a group and let a E G. An element bEG is called a conjugate of a if there exists an element x EG such that b-xax-¹. Show that any conjugate of a has the same order as a. 4.21 Show hat for any two elements x, y of any group G, o(xy)= o(yx). 4.22 Let G be an abelian group and let x, y E G. Suppose that x and y are of finite order. Show that xy is of finite order and that, in fact, o(xy) divides o(x)o(y). 4.23 Let G, x, y be as in Exercise 4.22, and assume in addition that (o(x), o(y))=1. Prove that o(xy)= o(x)o(y). 4.24 Let G be a group and let x, y E G. Assume that xe, o(y)=2, and yxy-¹=x². Find o(x).

4.14 Give an example of an infinite group G such that every element of G has finite order. 4.15 a) Find (123,321), and find integers x and y such that 123x+321y=(123,321). b) Find (862,347), and find integers x and y such that 862x+347y-(862,347). c) Find (7469,2464), and find integers x y such that 7469x+2464y= (7469,2464). 4.16 Prove that if G=(x), then G=(x¹). ted 4.17 Prove that if G=(x) and Gis inite, then x and x-are the only generators of G. 4.18 Prove parts (ii) and (iii) Theorem 4.1. 4.19 Prove part (i) of Theorem 4.4. 4.20 Let G be a group and let a E G. An element bEG is called a conjugate of a if there exists an element x EG such that b-xax-¹. Show that any conjugate of a has the same order as a. 4.21 Show hat for any two elements x, y of any group G, o(xy)= o(yx). 4.22 Let G be an abelian group and let x, y E G. Suppose that x and y are of finite order. Show that xy is of finite order and that, in fact, o(xy) divides o(x)o(y). 4.23 Let G, x, y be as in Exercise 4.22, and assume in addition that (o(x), o(y))=1. Prove that o(xy)= o(x)o(y). 4.24 Let G be a group and let x, y E G. Assume that xe, o(y)=2, and yxy-¹=x². Find o(x).

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.4: Cyclic Groups
Problem 11E: Exercises 11. According to Exercise of section, if is prime, the nonzero elements of form a...
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4.14 Give an example of an infinite group G such that every element of G has finite order. 4.15 a) Find (123,321), and find integers x and y such that 123x+321y=(123,321). b) Find (862,347), and find integers x and y such that 862x+347y-(862,347). c) Find (7469,2464), and find integers x y such that 7469x+2464y= (7469,2464). 4.16 Prove that if G=(x), then G=(x¹). ted 4.17 Prove that if G=(x) and Gis inite, then x and x-are the only generators of G. 4.18 Prove parts (ii) and (iii) Theorem 4.1. 4.19 Prove part (i) of Theorem 4.4. 4.20 Let G be a group and let a E G. An element bEG is called a conjugate of a if there exists an element x EG such that b-xax-1. Show that any conjugate of a has the same order as a. 4.21 Show hat for any two elements x, y of any group G, o(xy)= o(yx). 4.22 Let G be an abelian group and let x, y E G. Suppose that x and y are of finite order. Show that xy is of finite order and that, in fact, o(xy) divides o(x)o(y). 4.23 Let G, x, y be as in Exercise 4.22, and assume in addition that (o(x), o(y))=1. Prove that o(xy)= o(x)o(y). 4.24 Let G be a group and let x, y E G. Assume that xe, o(y)=2, and yxy-¹=x². Find o(x).
Transcribed Image Text:4.14 Give an example of an infinite group G such that every element of G has finite order. 4.15 a) Find (123,321), and find integers x and y such that 123x+321y=(123,321). b) Find (862,347), and find integers x and y such that 862x+347y-(862,347). c) Find (7469,2464), and find integers x y such that 7469x+2464y= (7469,2464). 4.16 Prove that if G=(x), then G=(x¹). ted 4.17 Prove that if G=(x) and Gis inite, then x and x-are the only generators of G. 4.18 Prove parts (ii) and (iii) Theorem 4.1. 4.19 Prove part (i) of Theorem 4.4. 4.20 Let G be a group and let a E G. An element bEG is called a conjugate of a if there exists an element x EG such that b-xax-1. Show that any conjugate of a has the same order as a. 4.21 Show hat for any two elements x, y of any group G, o(xy)= o(yx). 4.22 Let G be an abelian group and let x, y E G. Suppose that x and y are of finite order. Show that xy is of finite order and that, in fact, o(xy) divides o(x)o(y). 4.23 Let G, x, y be as in Exercise 4.22, and assume in addition that (o(x), o(y))=1. Prove that o(xy)= o(x)o(y). 4.24 Let G be a group and let x, y E G. Assume that xe, o(y)=2, and yxy-¹=x². Find o(x).
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