Question

Prove that a factor group of an Abelian group is Abelian.

   Prove that a factor group of an Abelian group is Abelian.

Contemporary Abstract Algebra

Contemporary Abstract Algebra

Joseph Gallian 9th Edition

Chapter 9, Problem 12

Instant Answer

Solved by Expert Nick Johnson

06/30/2023

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Let G be an Abelian group, which means that for any elements a, b in G, we have a * b = b * a. Show more…

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Prove that a factor group of an Abelian group is Abelian.

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Key Concepts

Normal Subgroup

A normal subgroup of a group is a subgroup that is invariant under conjugation by any element of the group, meaning for every element n in the subgroup and every group element g, the element g * n * g?¹ is still in the subgroup. This property is essential for forming factor groups, as it guarantees that the coset multiplication is well-defined.

Cosets

Cosets are the sets formed by multiplying all elements of a subgroup by a fixed element from the group. They partition the group into equivalence classes, and this partitioning is a key step in constructing factor groups. Cosets are integral to understanding the structure of groups and subgroups, particularly when discussing quotient structures.

Abelian Group

An Abelian group is a group in which the group operation is commutative; that is, for any two elements a and b in the group, the equation a * b = b * a holds. This property simplifies the structure and analysis of the group and is fundamental in many areas of algebra, ensuring that element interactions are symmetric.

Factor Group

A factor group (or quotient group) is formed by taking the set of cosets of a normal subgroup in a group and defining a group operation on these cosets. The operation is defined by the product of representatives from each coset. Factor groups help simplify group structures and are a central concept in many proofs and applications, including demonstrating properties like commutativity in the resulting group.

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