### Problem 10Let $ G $ be abelian and let $ H $ be a subgroup of $ G $. Show that $ G/H $ is abelian.#### ExplanationIn this problem, you're asked to show that the quotient group $ G/H $, formed by dividing the group $ G $ (which is abelian) by its subgroup $ H $, is also abelian. **Definitions:**- An **abelian group** is a group $ G $ where the operation is commutative, i.e., for any elements $ a, b $ in $ G $, the equation $ ab = ba $ holds. - A **subgroup** $ H $ of a group $ G $ is a subset of $ G $ that is itself a group under the operation of $ G $. - The **quotient group** $ G/H $ consists of the set of cosets of $ H $ in $ G $. Each coset is of the form $ gH $, where $ g $ is an element of $ G $.**Objective:**Your goal is to prove that the multiplication of cosets is commutative in the quotient group $ G/H $. More specifically, for any two elements $ aH $ and $ bH $ in $ G/H $, you need to show $ (aH)(bH) = (bH)(aH) $.In summary, the task is to demonstrate that the operation in the quotient group $ G/H $ maintains the commutative property if the parent group $ G $ is abelian.

We have to prove that given theorem.

Answered: 10. Let G be abelian and let H be a…

Math

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10. Let G be abelian and let H be a subgroup of G. Show that G/H is abelian.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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### Problem 10

Let $ G $ be abelian and let $ H $ be a subgroup of $ G $. Show that $ G/H $ is abelian.

#### Explanation

In this problem, you're asked to show that the quotient group $ G/H $, formed by dividing the group $ G $ (which is abelian) by its subgroup $ H $, is also abelian.

**Definitions:**

- An **abelian group** is a group $ G $ where the operation is commutative, i.e., for any elements $ a, b $ in $ G $, the equation $ ab = ba $ holds.

- A **subgroup** $ H $ of a group $ G $ is a subset of $ G $ that is itself a group under the operation of $ G $.

- The **quotient group** $ G/H $ consists of the set of cosets of $ H $ in $ G $. Each coset is of the form $ gH $, where $ g $ is an element of $ G $.

**Objective:**

Your goal is to prove that the multiplication of cosets is commutative in the quotient group $ G/H $. More specifically, for any two elements $ aH $ and $ bH $ in $ G/H $, you need to show $ (aH)(bH) = (bH)(aH) $.

In summary, the task is to demonstrate that the operation in the quotient group $ G/H $ maintains the commutative property if the parent group $ G $ is abelian.

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