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MTH376: Algebra

For

 Master of Mathematics

By

Dr. M. Fazeel AnwarAssistant Professor

Department of Mathematics, CIIT Islamabad

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Lecture 03

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 Recap

• Binary operation • Examples of binary operations • Motivation for defining groups• Group (Definition)

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Motivation for defining groups

Solution of linear equations

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Motivation (continued…)

Solve the following equation:

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Group

• A group  is a set together with a binary operation on such that the following axioms are satisfied:

1. The binary operation is associative.

2. There is an element such that for all

3. For each there is an element such that  for all

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Discussion and remarks

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Examples

• Lets look at some number sets first.

1) Is  a group? Yes No

2) Is  a group? Yes No

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Examples.

3) Is  a group? Yes No

4) Is  a group? Yes No

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Examples..

5) Is  a group? Yes No

6) Is  a group? Yes No

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Examples…

• Define   , .

7) Is a group? Yes No

8) Is a group? Yes No

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Examples….

• We will denote by the set of positive integers, positive rational numbers and positive real numbers respectively.

9) Is a group? Yes No

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Summary

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Thank You