## [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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This question was answered on: Sep 18, 2020

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