Homomorphism, Kernel and Coset

[LCSphysicist]

Homework Statement

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Relevant Equations

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I did the first step, that is, show that f is a homomorphism. Now i need to find the kernel K of f. But i am a little confused how to find it. Seeing the image, can we say the kernel is {0,4}?

 

[member 587159]

Homework Statement:: .
Relevant Equations:: .

View attachment 274584
I did the first step, that is, show that f is a homomorphism. Now i need to find the kernel K of f. But i am a little confused how to find it. Seeing the image, can we say the kernel is {0,4}?

Yes, you are correct. The kernel are all elements that are mapped to the identity of the group (0 here), and these elements are exactly 0 and 4.

 

[fresh_42]

It might be easier than checking $64$ possibilities to show that the modulo function, which assigns the remainder $r$ in a division by $n$ to $n$, and multiplication by $m$ are (additive) homomorphisms
$$
g\, : \,\mathbb{Z} \longrightarrow m\cdot \mathbb{Z}\; , \;z\longrightarrow m\cdot z
$$
$$
f\, : \,\mathbb{Z}\longrightarrow \mathbb{Z}_n\; , \; z\longmapsto r \text{ where } z=q\cdot n + r\, , \,0\leq r<n
$$