§ Semidirect product mnemonic

I just learnt that when we write the semidirect product NKN \ltimes K, the \ltimes is to look like a combination of NGN \triangleleft G ( NN is normal in GG) and the ×\times operator; This tells us that it is the NN part that is normal. in NKN \ltimes K.
  • A good example to remember is R3SO(3)\mathbb R^3 \ltimes SO(3), where we define a group element (t,r)(t, r) by the action: (t,r)v=rv+t(t, r) v = rv + t [ tt for translation and rr for rotation ].
  • Let's compose these. We find that:
(t2,r2)((t1,r1)(v))=(t2,r2)(r1v+t1)=r2(r1v+t1)+t2=(r2r1)v+(r2t1+t2)=(r2t1+t2,r221)v \begin{aligned} &(t2, r2)((t1, r1)(v)) \\ &= (t2, r2)(r1v + t1) \\ &= r2(r1v + t1) + t2\\ &= (r2 r1) v + (r2 t1 + t2) \\ &= (r2 t1 + t2, r2 21) v \end{aligned}
  • Here, we have the rotation r2r_2 act non-trivially on the translation t1t_1.
  • We need the translation to be normal, since we are messing with the translation by a rotation.
  • We want the translations to be closed under this messing about by the rotation action; The action of a rotation on a translation should give us another translation. Thus, the translations (t1,id)(t_1, id) ought to be normal in the full group (t2,r2)(t_2, r_2).
Another mnemonic for the semidirect product:
my thesis adviser told me that the acting group (the non-normal subgroup) opens its mouth and tries to swallow / "act on" the group it acts upon (the normal subgroup). The group that is acted on must be normal, because we act "by conjugation". Alternatively, being normal is "tasty", and thus needs to be eaten.