Is there a Quotient Group Isomorphic to C4 in Cyclic Group Order 16?

[Kate2010]

Homework Statement

I have to use the first isomorphism theorem to determine whether C16 (cyclic group order 16) has a quotient group isomorphic to C4.

Homework Equations

First isomorphism theorem

The Attempt at a Solution

C16 = {e, a, ..., a^15}
C4 = {e, b, ..., b^3}

Homomorphism f(a^m) = b^m 0<= m < 16
ker f is all x in C16 such that f(x) = e = b^4 = b^8 = b^12 = {e, a^4, a^8, a^12}
im f = {e, b^2, ..., b^15} = {e, b^2, b^3, b,..., b^3} = C4

Therefore, there is an isomorphism.

I'm unsure about my method here, especially finding I am f, as it initally appears that I am f is bigger than C4.

 

[Kate2010]

The next one I have to check is whether the subgroup A4 of S4 has a quotient group isomorphic to C4. My instinct says no but I have no idea how to prove this. Again, I'm advised to use the first isomorphism theorem.

 

[Tedjn]

Your method for the first question is fine. You are defining your function f from C₁₆ to C₄, so of course the image cannot be larger than C₄. All that remains is to make sure f is a homomorphism and onto. Both are true in your case.

For the second question, your instinct is correct. Here is a good way to think about this problem. List all 12 elements of A₄. What is the order of each of these elements? What does that imply about any homomorphism into C₄?

 

[Kate2010]

Ok, so besides the identity they all have order 2 or 3 I think. Whereas all elements besides the identity in C4 have order 4. But I don't think we can have a homomorphism that raises the order of elements?