i don't know what you mean by but yes, there are exactly 4 homomorphisms, because a homomorphism can send the generator of to any of 4 elements of

correct!

2)Describe explicitly all homomorphisms

h: C_5 ---> Aut(C_11)

Since 11 is prime number, so Aut(C_11) =~ C_10

I worked out that there are 5 homomorphism for C_5 ---> Aut(C_11). Is that true?

let where let then now can be defined to be any element of the set

3)This question I dont know how to do it.

Show that Aut(C_2*C_2) =~ D_6 ( from cyclic group to dihedral group)

Thank you in advanced for your time

since is one-to-one, can be defined to be any element of the set so there are 3 possibilities for and 2 possibilities for therefore

define the automorphisms by: and then and hence

is not cyclic. now use this fact that if is a prime number, then any group of order is either cyclic or it's isomorphisc to

alternatively, see that and [you need to check that all elements of are distinct.] but we have so: