This means we start with a set of elements and a rule for combining the elements. From there we say that in order to be a group, the following things must be true.The group is closed under its operation.The operation is associative.There is an identity element.Every element in the group has an inverse.Explain what each of these requirements and include an example to illustrate the concept. Why would each of these properties be important to include as part of the definition of a group?

Answered: This means we start with a set of…

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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This means we start with a set of elements and a rule for combining the elements. From there we say that in order to be a group, the following things must be true.

The group is closed under its operation.
The operation is associative.
There is an identity element.
Every element in the group has an inverse.

Explain what each of these requirements and include an example to illustrate the concept. Why would each of these properties be important to include as part of the definition of a group?

Expert Solution

Solution:

The group is closed under its operation.

Consider the set G and two elements a, b in it.

If the set G is a group is closed under its operation say addition, then $a + b \in G$ .

The operation is associative.

For elements a, b, c in G, $(a + b) + c = a + (b + c)$ .

There is an identity element e.

$a + e = a = e + a$

Every element a in the group has an inverse b.

$a + b = e = b + a$

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