Thread: Homomorphic images of D3.

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

Φ: D3 to Z2

Originally Posted by o&apartyrock

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

Φ: D3 to Z2

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

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