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## Homework Statement

Let, M={ (a -b) (b a):a,b∈ℝ}, show (H,+) is isomorphic as a binary structure to (C,+)

## Homework Equations

Isomorphism, Group Theory, Binary Operation

## The Attempt at a Solution

Let a,b,c,d∈ℝ

Define f : M→ℂ by f( (a -b) (b a) ) = a+bi

1-1:

Suppose f( (a -b) (b a) )= f( (c -d) (d c)), then a+bi =c+di, thus a=c and b=d, thus f is one to one.

Onto:

Let a+bi∈ℂ , then (a -b) (b a)∈M, so f((a -b) (b a))=a+bi, thus f is onto.

Homomorphic:

f((a -b) (b a) + (c -d ) (d c)))

=f( ((a+c) -(b+d)) ((b+d) -(a+c)))

= (a+c)+(b+d)i

= f((a -b) (b a)) +f((c -d) (d c))

I don't think I show f: M→ℂ is 1-1 and onto correctly because my instructor didn't talk about them to much, can any show me how show f: M→ℂ is 1-1 and onto? thanks!

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