## [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:
? closure property: For all r and t in S, r*t is also in S
? commutative property: For all r and t in S, r*t=t*r
? identity property: There exists an element e in S so that for every s in S, s*e=s
? inverse property: For every s in S, there exists an element x in S so that s*x=e
? 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:
1. closure property
2. commutative property
3. identity property
4. inverse property
5. associative property

First we find the 5th roots of unity. Let z 5 1 cos 0 i sin 0, as cos 0 1,sin 0 0

WE know that n distinct complex roots of z r cos i sin are given by the formula 2k

z k r1/ n cos n 2 k i sin where...

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