10.2.5. Let G be a group, let N < G, and let H ≤ G. Draw a partial latticediagram that includes G, N, H, NH, and H□N. Prove that H^N

FINAL ANSWER: (H∩N)is a normal subgroup of (H). Thank YouExplanation:STEP−1 To show that (H∩N) is…

Answered: 10.2.5. Let G be a group, let N < G,…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.7: Direct Sums (optional)

Problem 2E

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The elements of the multiplicative group G of 33 permutation matrices are given in Exercise 35 of section 3.1. Find the order of each element of the group. (Sec. 3.1,35) A permutation matrix is a matrix that can be obtained from an identity matrix In by interchanging the rows one or more times (that is, by permuting the rows). For n=3 the permutation matrices are I3 and the five matrices. (Sec. 3.3,22c,32c, Sec. 3.4,5, Sec. 4.2,6) P1=[ 100001010 ] P2=[ 010100001 ] P3=[ 010001100 ] P4=[ 001010100 ] P5=[ 001100010 ] Given that G={ I3,P1,P2,P3,P4,P5 } is a group of order 6 with respect to matrix multiplication, write out a multiplication table for G.
4.6 List all the elements of (Z45, ) that are of order 15.
5.1 In each case, determine whether or not H is a subgroup of G. a) G=(R, +); H=Q b) G=(Q, +); H=Z c) G=(Z, +); H=Z* d) G=(Q-(0}, ); H=Q+ e) G=(Zg, Ð); H=(0,2,4} f) G=the set of 2-tuples of real numbers (a,b) under addition of 2-tuples; H=the subset consisting of all 2-tuples such that b=-a g) G=Qs; H={I, J, K} h) G=(P(X), A); H={Ø, A, B, A AB}, where A, B are two elements of G i) G=(P(X), A); H=P(Y), where YCX.
3. Give the lattice of subgroups of Z₂ X Z₁.
10.2.11. Let G = D34 = (a,b | a¹7 = b² = e,ba = a¹6b). Let H = (a) and K = = (b). (a) Draw a partial lattice diagram of subgroups of G that includes the subgroups H and K. Include also H¦ K and (H,K). Label the edges with appropriate numbers, and give reasons for what you have done. In what follows, you may refer back to your diagram. (b) Is H
8. Let (G, *) be a group. Define the center of G by Z(G) := {x € G: x * a = a * x, Va E G}. The set Z(G) consists of all elements of G that commute with every possible element of the group. For example, one can say that the matrix 41 belongs to the center of (GL(2, R), ·) because (4I)A = A(4I) for all A in GL(2, R), since both sides are equal to 4A. (a) Show that, for every group G, the center Z(G) is a subgroup of G. (b) Find the center of (Z4, +) and (this part is moved to next homework) the center of D6, the dihedral group. (You should be able to tell from the group table.) (c) One could say that "the center Z(G) measures the abelian-ness of a group G". Please interpret this statement. 'Recall that a group (G, *) is called abelian if the operation * is commutative. Hint: What is Z(G) equal to when G is abelian?
7. Consider an element z = the order of r is the least common multiple of the orders of its components a,. ([as),. (1,) in the product group Z/n,Zx ... x Z/n,Z. Show that ..... 9.
4, .3 Let G=U(18) and H={1,7,13} be a subgroup of G. The number of distinct left cosets of H in G is * O 2
The group (U20, X20) = {(1, 3, 7, 9, 11, 13, 17, 19), X20} has subgroups H = {1,3,7,9) and K = {1,11}. qergie laurion Sonnen (a) Explain why both H and K are normal subgroups of U20, and for each of H and K list all of its distinct cosets in U20. (b) For each of H and K, write down the group table of its quotient group in U20, and state a standard group to which the quotient group is isomorphic, briefly justifying your answer. 1010 (The standard groups introduced in M208 are listed on pages 136-137 of the Handbook.) 10% 10 vlad 1990s a alwallal way soilitur
Determine if the following sets with operations are groups: (a) R with the operation given by r * y = -1 - y. (b) The subset of R® given by H = {(r,y, 2) € R* |x+ y+z = 0}, with the operation (x, y, 2) + (a, b, c) = (r +a, y +b, z + c).
10.1.9. Find a group G, with subgroups H and K, such that H◄K, K◄G, but H not normal in G.
2. List all the elements of (Z30, Đ) that are of order 15.

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10.2.5. Let G be a group, let N < G, and let H ≤ G. Draw a partial lattice diagram that includes G, N, H, NH, and H□N. Prove that H^N