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## 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.