Question

What is the number of group homomorphisms from z12 to z13?

Answer 1

To find homomorphisms from Z12 to Z13.

Think Z12 is cyclic group of order 12

Let 'a' be the generator of Z12.

Clearly o(a)=12

O(Im(a) )/o(a) i.e. O(I'm(a)) can be 1,2,3,4,6,12.

But Z13 does not have any elements of order 2,3,4,6 and 12 as these numbers does not divide o(Z13)=13.

So O(Im(a)) =1.

This implies 'a' maps to ‘e' identity of Z13.

Which is trivial homomorphism.

Hence only one homomorphism possible.

Number of homomorphism from Zm to Zn is g.c.d(m,n).

Here, g.c.d(12,13)=1