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Groups: Handwritten notes by Atiq ur Rehman http://www.MathCity.org/atiq Partial Contents 1. Groups; definition and examples 1 2. Order of group, order of element 3 3. Periodic group, mixed group 4 4. Subgroup 6 5. Invalution 8 6. Relation between groups, homomorphism, monomorphism, epimorphism, isomorphism, endomorphism, examples and related theorem 9 7. Kernel, definition and related theorems 13 8. Cyclic group, related theorems 18 9. Complex in a group, product of complexes and related theorems 24 10. Coset, definition and examples 30 11. Index of subgroup, Lagrange’s theorem 31 12. Double coset, related theorem 33 13. Normalizer, definition and related theorems 34 14. Centralizer, centre of group, related theorem 36 15. Conjugate or transform of a group, definition and related theorems 37 a

Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Page 1: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

Groups: Handwritten notes

by

Atiq ur Rehman

http://www.MathCity.org/atiq

Partial Contents

1. Groups; definition and examples 1

2. Order of group, order of element 3

3. Periodic group, mixed group 4

4. Subgroup 6

5. Invalution 8

6. Relation between groups, homomorphism, monomorphism, epimorphism, isomorphism,

endomorphism, examples and related theorem 9

7. Kernel, definition and related theorems 13

8. Cyclic group, related theorems 18

9. Complex in a group, product of complexes and related theorems 24

10. Coset, definition and examples 30

11. Index of subgroup, Lagrange’s theorem 31

12. Double coset, related theorem 33

13. Normalizer, definition and related theorems 34

14. Centralizer, centre of group, related theorem 36

15. Conjugate or transform of a group, definition and related theorems 37

a

Page 2: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

16. Self conjugate, conjugancy class, related theorem 40

17. Class equation, p-group, definition and related theorems 43

18. Conjugate subgroup, definition and related theorems 45

19. Normal subgroup, definition and related theorems 49

20. Factor or quotient group, definition and related theorem 53

21. 1st isomorphism theorem, related theorem 55

22. 2nd isomorphism theorem 58

23. 3rd isomorphism theorem 60

24. Endomorphism, automorphism, definition and related theorem 62

25. Conjugation as an automorphism 64

26. Inner and outer automorphism, definition and related theorems 65

27. Commutator of a group, definition and related theorem 70

28. Derive group or commutative group, definition and related theorem 71

29. Direct product of groups, definition and related theorems 73

30. Invariant subgroup 79

31. Characteristic subgroup 80‘

Available at http://www.MathCity.org/msc

If you have any question, ask at http://forum.mathcity.org

MathCity.org is a non-profit organization, working to promote mathematics in Pakistan. If

you have anything (notes, model paper, old paper etc.) to share with other peoples, you

can send us to publish on MathCity.org. You may earn money by participating. For more

information visit: http://www.MathCity.org/participate.html

b

Page 3: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

[1]

MathCity.org Merging man and maths

Groups (Handwritten notes) Written by: Atiq ur ReWritten by: Atiq ur ReWritten by: Atiq ur ReWritten by: Atiq ur Rehman, Class: BS or MSc (Maths)hman, Class: BS or MSc (Maths)hman, Class: BS or MSc (Maths)hman, Class: BS or MSc (Maths)

Available online at http://www.MathCity.org , Version: 1.0.0

Page 4: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

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Available online at http://www.MathCity.org **********************************************

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Groups: Handwritten notes

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Page 6: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

Page 7: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

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Groups: Handwritten notes

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Groups: Handwritten notes

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Groups: Handwritten notes

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Groups: Handwritten notes

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Available online at http://www.MathCity.org

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Groups: Handwritten notes

Page 22: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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[20]

Groups: Handwritten notes

Page 23: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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[21]

Groups: Handwritten notes

Page 24: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

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Available online at http://www.MathCity.org

**********************************************

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Groups: Handwritten notes

Page 26: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

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Available online at http://www.MathCity.org

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Page 27: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

Page 28: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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[26]

Groups: Handwritten notes

Page 29: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

**********************************************

Available online at http://www.MathCity.org

**********************************************

Page 30: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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[29]

Groups: Handwritten notes

**********************************************

Available online at http://www.MathCity.org

**********************************************

Page 32: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

Page 33: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

Page 34: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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[32]

Groups: Handwritten notes

**********************************************

Available online at http://www.MathCity.org

**********************************************

Page 35: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

Page 36: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

Page 37: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

**********************************************

Available online at http://www.MathCity.org

**********************************************

Page 38: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

Page 39: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

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Groups: Handwritten notes

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Groups: Handwritten notes

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Groups: Handwritten notes

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Groups: Handwritten notes

**********************************************

Available online at http://www.MathCity.org

**********************************************

Page 48: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

Page 56: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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[55]

Groups: Handwritten notes

Page 58: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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[56]

Groups: Handwritten notes

**********************************************

Available online at http://www.MathCity.org

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Discuss your problems at http://forum.mathcity.org

******************************************

Page 59: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

**********************************************

Available online at http://www.MathCity.org

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Page 60: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

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[60]

Groups: Handwritten notes

Page 63: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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[61]

Groups: Handwritten notes

**********************************************

Available online at http://www.MathCity.org

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Page 64: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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[62]

Groups: Handwritten notes

Page 65: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

Page 66: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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[64]

Groups: Handwritten notes

Page 67: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

Page 68: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

**********************************************

Available online at http://www.MathCity.org

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***************************************************************

� FSc

� BSc

� MSc / BS

� MPhil / MS

� PhD

� Old Papers / Entry Test

o Check out all these at

� http://www.MathCity.org

***************************************************************

Page 69: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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[68]

Groups: Handwritten notes

**********************************************

Available online at http://www.MathCity.org

**********************************************

Ask questions at http://forum.mathcity.org

**********************************************

***************************************************************

� FSc

� BSc

� MSc / BS

� MPhil / MS

� PhD

� Old Papers / Entry Test

o Check out all these at

� http://www.MathCity.org

***************************************************************

Page 71: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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Groups: Handwritten notes

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Check out notes of other subjects at http://www.MathCity.org

*****************************************************

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[75]

Groups: Handwritten notes

***************************************************************

� FSc

� BSc

� MSc / BS

� MPhil / MS

� PhD

� Old Papers / Entry Test

o Check out all these at

� http://www.MathCity.org

***************************************************************

Page 78: Atiq ur Rehman - MathCity.org 16. Self conjugate, conjugancy class, related theorem 40 17. Class equation, p-group, de nition and related theorems 43 18. Conjugate subgroup, de nition

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={ The End }=

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