Homomorphic images of D3.

**o&apartyrock** · Apr 28th 2009, 04:38 PM

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

Φ: D3 to Z2

**NonCommAlg** · Apr 28th 2009, 04:55 PM

Originally Posted by o&apartyrock

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

Φ: D3 to Z2

we have $D_3=<a,b: \ a^2=b^3=1, \ ab=b^{-1}a>=\{1,a,b,b^2,ab,ab^2 \}.$ 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 $<b>.$ so we must have $f(1)=f(b)=f(b^2)=\bar{0}, \ f(a)=f(ab)=f(ab^2)=\bar{1}.$

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