Exploring Homomorphisms and Automorphisms for Cyclic Groups

[MidnightR]

Describe explicitly all homomorphisms

h: C_6 ----> Aut(C_n)

The question asks when n=12,16

I was wondering if someone could explain how to do this? I've looked through the notes but struggling a tad

I think I could do this if it said for instance h: C_6 ----> C_n but Aut(C_12) = C_2 x C_2 which is a little confusing

Thanks

 

[Hurkyl]

Can you articulate why you think finding homomorphisms

C_6 ---> C_2 x C_2​

is hard and confusing, but finding homomorphisms

C_6 ---> C_n​

is not?

 

[MidnightR]

Can you articulate why you think finding homomorphisms

C_6 ---> C_2 x C_2​

is hard and confusing, but finding homomorphisms

C_6 ---> C_n​

is not?

Lack of understanding & having an example for the latter but not the former xD

I think I've figured it out though

for n = 12 there are 4 homomorphisms

h_1(x) = 1
h_2(x) = z
h_3(x) = y
h_4(x) = zy

for n = 16 there are 8 homomorphisms

h_1(x) = 1
h_2(x) = z
h_3(x) = z^2
h_4(x) = z^3
h_5(x) = y
h_6(x) = zy
h_7(x) = z^2y
h_8(x) = z^3y

Would that be correct?

For the next question all homomorphism h: C_5 --> Aut(C_11) I get 5

h_1(x) = 1
h_2(x) = z^2
h_3(x) = z^4
h_4(x) = z^6
h_5(x) = z^8

 

[Hurkyl]

Can you say why you think that's the answer? Why must any homomorphism be named on that list, and why is everything named on that list a homomorphism?