3. We showed in class that every subgroup of a cyclic group is cyclic. Let n e N.(a) An integer combination of two numbers m and n is a number of the form mx+ny, wherex, y E Z. A result in number theory (see Pinter p. 219) says that a number a is an integercombination of m and n if and only if a is a multiple of gcd(m, n) (and as a consequence,gcd(m, n) is the smallest positive integer combination of m and n). Use this to provethat if m e Zn, then (m) = (gcd(m, n)). (Hint: xm = xm + ny (mod n).)(b) Use this to fill in details of the result in class that if m e Zn, then ord(m)gcd(m,n)*(c) Prove that for every divisor d of n, Zn has a unique subgroup of order d. (Hint: What isthe order of the element ?)

C) Let k be a subgroup of order d, then k is cyclic and generated by an element of order k =K⊂H…

Answered: (c) Prove that for every divisor d of…

(c) Prove that for every divisor d of n, Zn has a unique subgroup of order d. (Hint: What is the order of the element "?)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.4: Cosets Of A Subgroup

Problem 32E: (See Exercise 31.) Suppose G is a group that is transitive on 1,2,...,n, and let ki be the subgroup...

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Question

3. We showed in class that every subgroup of a cyclic group is cyclic. Let n e N.
(a) An integer combination of two numbers m and n is a number of the form mx+ny, where
x, y E Z. A result in number theory (see Pinter p. 219) says that a number a is an integer
combination of m and n if and only if a is a multiple of gcd(m, n) (and as a consequence,
gcd(m, n) is the smallest positive integer combination of m and n). Use this to prove
that if m e Zn, then (m) = (gcd(m, n)). (Hint: xm = xm + ny (mod n).)
(b) Use this to fill in details of the result in class that if m e Zn, then ord(m)
gcd(m,n)*
(c) Prove that for every divisor d of n, Zn has a unique subgroup of order d. (Hint: What is
the order of the element ?)

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