PROBLEM 2 2.1 Let n ≥ 3 and let H = (r") C D2n. Show that H C Z(D2n). Is H a normal subgroup? What is the order of the quotient set D2n/H? 2.2 Let G be a group and let H C G be nonempty. Write down the definitions of CG(H) and NG(H). Is it true that NG (H) ℃ Cg(H)? 2.3 Give an example of a group which has one subgroup of order k for each k = {1, 2, 4, 8, 16}, and no other subgroups. 2.4 Write down the statement of the First Isomorphism Theorem.

PROBLEM 2 2.1 Let n ≥ 3 and let H = (r") C D2n. Show that H C Z(D2n). Is H a normal subgroup? What is the order of the quotient set D2n/H? 2.2 Let G be a group and let H C G be nonempty. Write down the definitions of CG(H) and NG(H). Is it true that NG (H) ℃ Cg(H)? 2.3 Give an example of a group which has one subgroup of order k for each k = {1, 2, 4, 8, 16}, and no other subgroups. 2.4 Write down the statement of the First Isomorphism Theorem.

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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