21. Let G be an abelian group with subgroup H. Let G/H be the set of cosets of H in G. Define multiplication of congruence classes by aH-bH = abH. Prove that if aH = a'H and bH = b'H, then abH = a'b'H, and so multiplication of cosets is well-defined. Prove that G/H is an abelian group with this multiplication. This is called the quotient group of G by H.
21. Let G be an abelian group with subgroup H. Let G/H be the set of cosets of H in G. Define multiplication of congruence classes by aH-bH = abH. Prove that if aH = a'H and bH = b'H, then abH = a'b'H, and so multiplication of cosets is well-defined. Prove that G/H is an abelian group with this multiplication. This is called the quotient group of G by H.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.6: Quotient Groups
Problem 32E: 32. Let be a fixed element of the group . According to Exercise 20 of section 3.5, the mapping ...
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![21. Let G be an abelian group with subgroup H. Let G/H be the set of
cosets of H in G. Define multiplication of congruence classes by
aH.bH = abH.
Prove that if aH = a'H and bH = b'H, then abH = a'b'H, and so
multiplication of cosets is well-defined. Prove that G/H is an abelian
group with this multiplication. This is called the quotient group of G
by H.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F30761ad5-6d22-4ff4-adb6-4f166a7ab52a%2Fef66d38e-6158-4a44-8e77-d8b59672f721%2Fu8aq23_processed.jpeg&w=3840&q=75)
Transcribed Image Text:21. Let G be an abelian group with subgroup H. Let G/H be the set of
cosets of H in G. Define multiplication of congruence classes by
aH.bH = abH.
Prove that if aH = a'H and bH = b'H, then abH = a'b'H, and so
multiplication of cosets is well-defined. Prove that G/H is an abelian
group with this multiplication. This is called the quotient group of G
by H.
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