Proving Isomorphism of Z4 / (2Z4) and Z2

[DanielThrice]

Homework Statement

Why does it make sense (when considering Z4)to form the factor group

Z4 / (2Z4) where kZn = {0, k mod n, 2k mod n, ..., nk mod n}?

I believe that this above factor group is isomorphic to Z2, but how can I prove this?

 

[Tedjn]

The quotient group exists because 2Z4 is a normal subgroup of Z4 (since Z4 is abelian, all its subgroups are normal). To show that it is isomorphic to Z2, list the elements of Z4/2Z4 (which are cosets), and write down the isomorphism. Then, prove what you have is indeed an isomorphism, i.e. that it has the properties of an isomorphism.

 

[JonF]

How many elements are in Z4/2Z4? How many groups have that many elements?