Question

Show that set of all non zero complex numbers C* forms abelian group under multiplication of complex number.​

Answer 1

Answer:

Let C≠0 be the set of complex numbers without zero, that is:

C≠0=C∖{0}

The structure (C≠0,×) is an infinite abelian group.

Proof

Taking the group axioms in turn:

G0: Closure

Non-Zero Complex Numbers Closed under Multiplication.

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G1: Associativity

Complex Multiplication is Associative.

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G2: Identity

From Complex Multiplication Identity is One, the identity element of (C≠0,×) is the complex number 1+0i.

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G3: Inverses

From Inverse for Complex Multiplication‎, the inverse of x+iy∈(C≠0,×) is:

1z=x−iyx2+y2=z¯¯¯zz¯¯¯

where z¯¯¯ is the complex conjugate of z.

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C: Commutativity

Complex Multiplication is Commutative.

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Infinite

Complex Numbers are Uncountable.

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Step-by-step explanation: