# First isomorphism theorem

## 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 im f, as it initally appears that im f is bigger than C4.

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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.

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?

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?