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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?