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[answered] A set S with the operation * is an Abelian group if the fol


  1. A?set?S?with?the?operation?*?is?an?Abelian?group?if?the?following?five?properties?are?shown?to?be?true:
  2. ??closure?property:?For?all?r?and?t?in?S,?r*t?is?also?in?S
  3. ??commutative?property:?For?all?r?and?t?in?S,?r*t=t*r
  4. ??identity?property:?There?exists?an?element?e?in?S?so?that?for?every?s?in?S,?s*e=s
  5. ??inverse?property:?For?every?s?in?S,?there?exists?an?element?x?in?S?so?that?s*x=e
  6. ??associative?property:?For?every?q,?r,?and?t?in?S,?q*(r*t)=(q*r)*t

    A.?Prove?that?the?set?G?(the?fifth?roots?of?unity)?is?an?Abelian?group?under?the?operation?*?(complex?multiplication)?by?using?the?definition?given?above?to?prove?the?following?are?true
  7. :1.?closure?property
  8. 2.?commutative?property
  9. 3.?identity?property
  10. 4.?inverse?propert
  11. y5.?associative?property

A cayley table is not sufficient enough to prove the associative property. I need detailed step by step equations with defined justification.


The five roots:

 

2? ?0

 

2 ??0

 

0

 

0

 

cos

 

+i sin

 

cos + isin = 5

 

5

 

5

 

5 = 1+ 0

 

2 ??1

 

2 ? ?1

 

+ isin

 

=?

 

2?

 

2?

 

5

 

5 cos 5 +i sin 5 cos ?

 

2 ??2

 

2 ? ?2

 

4?

 

4?

 

+isin

 

=? cos

 

+i sin

 

5

 

5

 

5

 

5...

 


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