# Math Help - Homomorphic images of D3.

1. ## Homomorphic images of D3.

If Z2 is a homomorphic image of D3, what is the homomorphism?

Φ: D3 to Z2

2. Originally Posted by o&apartyrock

If Z2 is a homomorphic image of D3, what is the homomorphism?

Φ: D3 to Z2
we have $D_3==\{1,a,b,b^2,ab,ab^2 \}.$ let $f: D_3 \to \mathbb{Z}/2$ be an onto homomorphism. then we must have $|\ker f|=3.$

the only normal subgroup of $D_3$ of order 3 is $.$ so we must have $f(1)=f(b)=f(b^2)=\bar{0}, \ f(a)=f(ab)=f(ab^2)=\bar{1}.$