§ Semidirect product as commuting conditions

Recall that in NK=GN \ltimes K = G, NN is normal. This is from the mnemonic that it looks like NGN \triangleleft G, or from the fact that the acting/twisting subgroup KK is a fish that wants to "eat"/act on the normal subgroup NN. So, we have knk1Nknk^{-1} \in N as NN is normal, thus knk1=nknk^{-1} = n'. This can be written as kn=nkkn = n'k. So:
  • When commuting, the element that gets changed/twisted in the normal subgroup.This is because the normal subgroup has the requisite constraint on it to betwistable.
  • The element that remains invariant is the actor.
In the case of translations and rotations, it's the translations that are normal. This can be seen either by noticing that they are abelian, and are thus normal, while rotations don't "look normal". Alternatively, one can try to consider translate-rotate versus rotate-translate.
  • First rotating by rr and then translating by tt along the x-axishas the same effect as first translating by tt' at 45 degrees to the x-axis,and then rotating by the same r.
  • This begs the question, is there some othertranslation t'' and some other rotation r'' such that t''; r'' (t'' first, r'' next)has the same effect as r;t (r first, t next)?
  • First let's translate by tt along the x-axis and then rotating by rr.Now let's think, if we wanted to rotate and then translate, what rotation wouldwe start with? it would HAVE TO BE rr, since there's no other way to get the axisin such an angle in the final state. But if we rotate by rr, then NO translationcan get us to the final state we want. So, it's impossible to find a rotation;translationpair that mimics our starting translation;rotation.