(adsbygoogle = window.adsbygoogle || []).push({}); 1. I need to show that the quotient group G/ℝ+ is isomorphic to U using the Fundamental Theorem on group homomorphisms. I believe I can get there if someone can just help me find what ψ, the homomorphism function is supposed to be.

G=(ℂ-{0},*)

U={x+yi|x^2+y^2=1}

2. Relevant equations

Fundamental Theorem on group homomorphisms

3. The attempt at a solution

Note that ℝ⁺ is the subset of G where for any (x+yi)∈G, y=0.

First, we need to find a homomorphism ϕ:G→U such that ker(ϕ)=ℝ⁺.

Need to find: a,b such that (x+yi)⋅(a+bi)=(x+yi) given x²+y²=1 and a²+b²=1.

(x+yi)⋅(a+bi)=ax+bxi+ayi-by=(ax-by)+(bx+ay)i=(x+yi)

We also know (ax-by)²+(bx+ay)²=1.

Then (ax-by)=x and (bx+ay)=y.

So, a=1 and b=0, so e_{U}=1+0i.

Therefore, ϕ(x+yi)=1+yi, since only y=0 will give you 1+0i, which is e_{U}, which makes R⁺ be the ker(ϕ).

This ϕ is not onto.

ϕ(x+yi) is a homomorphism since for any a+bi,c+di∈G, ϕ((a+bi)⋅(c+di))=ϕ((ac-bd)+(ad+bc)i)=1+(ad+bc)i≠***(1-bd)+(b+d)i=(1+bi)⋅(1+di)=ϕ(a+bi)⋅ϕ(c+di). ***this is not equal

Then, by Thm 13.2 (Fundamental Theorem on group homomorphisms), ϕ(G)=G/ker(ϕ). So, (1+yi)≅G/R⁺.

****I went wrong somewhere since it's supposed to be U, not (1+yi).

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# I need to show that (C-{0},*)/{x+yi|x^2+y^2=1} is isomorphic to (R+,*)

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