Your proof of the homomorphism property is fine. Your description of the image is also correct but you should be able to describe it as something familiar.
There's more to the kernel.
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For real exponents, , but this isn't true for complex numbers.
Let be the map . Show that is a homomorphism and determine its image and kernal.
Okay, so this seems somewhat easy, but I am fairly new to this concept and groups in general, so I want to check if my work is correct.
Proof: Let . Then , so is a homomorphism. Q.E.D.
Am I allowed to distribute in the exponent? I am not sure since we are talking about groups with different laws of composition and no mentioning of distribution or anything, so any feedback on that would be appreciated.
As for the image, I think it's just the set for some .
As for the kernel, since the identity of is 1, we have that whenever , or when , so the kernel of is .
Again, any feedback/critique would be appreciated, thanks.