Homomorphisms and kernels,images

[kathrynag]

Homework Statement

Show that the function i : Z12 → Z12 defined by i([a]) = 3[a] for all [a] ∈ Z12 is a
group homomorphism and determine the kernel and image.

Homework Equations

The Attempt at a Solution

Well, I started by computing i([a]i()
=3[a]3
=9[ab]
It should equal i[ab], but that equals 3[ab]

 

[annoymage]

Z12 are group on operation addition, not multiplication

 

[kathrynag]

Oh that makes a lot more sense now.
kernel you want i(a)=e
kernel= [9]
I'm not quite sure about image.

 

[Outlined]

[0] is in the kernel too, also [4] and [8] and maybe even more

 

[kathrynag]

[0],[4],[[8]
Now that I think about [9] isn't in the kernel 3[9]=[27]=[3], not e

 

[Outlined]

Well, it seems that we found the kernel as [10] and [11] are NOT in the kernel.

 

[kathrynag]

yeah [10] and [11] are not in the kernel.
3[10]=[30]=6
3[11]=33=11