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## Homework Statement

Are these functions homomorphisms, determine the kernel and image, and identify the quotient group up to isomorphism?

C^∗ is the group of non-zero complex numbers under multiplication, and C is the group of all complex numbers under addition.

## Homework Equations

φ1 : C−→C

z −→ (Re(z))^2;

φ2 : C−→C

z −→ z^macron (conjugate of z) + iz;

φ3 : C^∗ −→ C^∗

z −→ (z^macron (conjugate of z))^2;

φ4 : C∗ −→ C∗

z −→ i/z

## The Attempt at a Solution

I found elements in each of φ1 and φ4 to show they are not homomorphisms.

In φ2, I find that φ(z1 + z2) = (z1^macron + iz1) + (z2^macron + iz2) = φ(z1) + φ(z2), hence a homomorphism.

Identity element under addition is zero, hence Ker(φ) is z^macron + iz = 0, so z^macron = -iz. Not sure if this is correct. So φ is onto and the image is C, and not sure of quotient group?

In φ3, I find that φ(z1z2) = φ(z1)φ(z2), hence a homomorphism.

Identity element under multiplication is 1, Ker(φ) is Z^macron = 1. So φ is onto and the image is C^∗ , and not sure of quotient group?