2(c)(d) 2. It is relevant in number theory to understand the group structure of Z. In thisproblem we will work out the structure of Z when n = 2m. The cases m = 1 andm = 2 are easy to compute directly: we have Z {1} is the trivial group, andZ₁ = {1,3} ≈ Z₂. So we suppose m ≥ 3, and let G = Zm.(a) Use mathematical induction to show that for every integer k ≥ 0, we have52%:= 1+ 2+2qfor some odd integer q.(b) Show that the order of 5 in G is 2m-2(c) Consider the subgroups H==1) and K = (5) of G. Show thatG≈H×K≈ Z₂ × Z2m-21000(d) As an application to number theory, find the number of positive integers x lessthan 128 such that x 1 is divisible by 128. (Hint: Show that this is thesame as the number of elements (a, b) € Z2 × Z25 such that (1000 ā, 1000 6)(0,0).)=

Lot of number theory facts will be used to solve this problem.

Answered: (c) Consider the subgroups H = (−1) and…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.8: Some Results On Finite Abelian Groups (optional)

Problem 14E: Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic...

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In Exercises 3 and 4, let G be the octic group D4=e,,2,3,,,, in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group D4 of rigid motions of a square The elements of the group D4 are as follows: 1. the identity mapping e=(1) 2. the counterclockwise rotation =(1,2,3,4) through 900 about the center O 3. the counterclockwise rotation 2=(1,3)(2,4) through 1800 about the center O 4. the counterclockwise rotation 3=(1,4,3,2) through 2700 about the center O 5. the reflection =(1,4)(2,3) about the horizontal line h 6. the reflection =(2,4) about the diagonal d1 7. the reflection =(1,2)(3,4) about the vertical line v 8. the reflection =(1,3) about the diagonal d2. The dihedral group D4=e,,2,3,,,, of rigid motions of the square is also known as the octic group. The multiplication table for D4 is requested in Exercise 20 of this section.
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