(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let H be a finite subgroup of a group G. Verify that the formula (h,h')(x)=hxh'^{-1}defines an action of H x H on G. Prove that H is a normal subgroup of G if and only if every orbit of this action contains precisely |H| points.

3. The attempt at a solution

I solved the first part of the question:

[tex]\left(\left(h,h'\right)\left(k,k'\right)\right)\left(x\right)=\left(hk,h'k'\right)\left(x\right)=hkx\left(h'k'\right)^{-1}=hkxk'^{-1}h'^{-1}=\left(h,h'\right)\left(\left(k,k'\right)\left(x\right)\right)[/tex]

This shows that the formula is a group homomorphism from H x H to G, and therefore it defines an action. But for the second part of the question I need a hint.

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# The size of the orbits of a finite normal subgroup

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