Problem 1. In each of the following, either prove or disprove that the given subsetHis a subgroup of G.1.1. G = (Q,+) and H = { : a, d = Z with d\n}, where n € Z+ is fixed.1.2. GS10, and H = {σ = S10 : |σ| = 4} U {e}.1.3. G = GL2(R) andH={ ( a ) : a: a, b, c € R with ac 40}.

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Answered: Problem 1. In each of the following,…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.5: Normal Subgroups

Problem 29E: Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.

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22. If and are both normal subgroups of , prove that is a normal subgroup of .
Let G be the group and H the subgroup given in each of the following exercises of Section 4.4. In each case, is H normal in G? Exercise 3 b. Exercise 4 c. Exercise 5 d. Exercise 6 e. Exercise 7 f. Exercise 8 Section 4.4 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. 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. 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. Let H be the subgroup (1),(2,3) of S3. Find the distinct left cosets of H in S3, write out their elements, partition S3 into left cosets of H, and give [S3:H]. Find the distinct right cosets of H in S3, write out their elements, and partition S3 into right cosets of H. In Exercises 7 and 8, let G be the multiplicative group of permutation matrices I3,P3,P32,P1,P4,P2 in Example 6 of Section 3.5 Let H be the subgroup of G given by H=I3,P4={ (100010001),(001010100) }. Find the distinct left cosets of H in G, write out their elements, partition G into left cosets of H, and give [G:H]. Find the distinct right cosets of H in G, write out their elements, and partition G into right cosets of H. Let H be the subgroup of G given by H=I3,P3,P32={ (100010001),(010001100),(001100010) }. Find the distinct left cosets of H in G, write out their elements, partition G into left cosets of H, and give [G:H]. Find the distinct right cosets of H in G, write out their elements, and partition G into right cosets of H.
Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?
For each of the following subgroups H of the addition groups Z18, find the distinct left cosets of H in Z18, partition Z18 into left cosets of H, and state the index [ Z18:H ] of H in Z18. H= [ 8 ] .
Please, don't just write in words. solve the questions, please.
10.1.9. Find a group G, with subgroups H and K, such that H◄K, K◄G, but H not normal in G.
6.14. Let π E SÅ be the permutation defined by TT (2) = 8, T(1) = 3, π(5) = 6, TT (6) = 1, TT (3) = 5, π(7) = 7, T(4) = 4, π(8) = 2. (a) Prove that ☛ has order 4. Let G be the subgroup for S8 generated by ; i.e., G {е, π, T², T³}. = = (π) = (b) Describe all of the orbits of G, as was done in Example 6.18. (c) Let X {1, 2, 3, 4, 5, 6, 7, 8}, so G acts on X. For each k € X, describe the stabilizer Gk. (d) Use your data from (b) and (c) to explicitly verify the orbit-stabilizer counting formula in Theorem 6.21. See (6.9) in Example 6.22 for a similar example.
12.
4. Let H be a normal subgroup of Sn. (a) Show that if H contains a 2-cycle then H = Sn. (b) Show that if H contains a 3-cycle then H = An or Sn. (c) Find all normal subgroups of S4.
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.
Ff.125.
10.3.3. Let G = Z/4Z × Z/6Z, and let H be the subgroup of G generated by (2,2). (a) What are the elements of H? (b) What are the elements of G/H? (c) Find a familiar group that is isomorphic to G/H.

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Problem 1. In each of the following, either prove or disprove that the given subset His a subgroup of G. 1.1. G = (Q,+) and H = { : a, d = Z with d\n}, where n € Z+ is fixed. 1.2. GS10, and H = {σ = S10 : |σ| = 4} U {e}. 1.3. G = GL2(R) and H = { ( a ) : a : a, b, c € R with ac 40}.