A Universe of Sorts

§ Semidirect product mnemonic

I just learnt that when we write the semidirect product

N \ltimes K

, the

\ltimes

is to look like a combination of

N \triangleleft G

(

N

is normal in

G

) and the

\times

operator; This tells us that it is the

N

part that is normal. in

N \ltimes K

A good example to remember is $\mathbb R^3 \ltimes SO(3)$ , where we define a group element $(t, r)$ by the action: $(t, r) v = rv + t$ [ $t$ for translation and $r$ for rotation ].

Let's compose these. We find that:

\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 $r_2$ act non-trivially on the translation $t_1$ .
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 $(t_1, id)$ ought to be normal in the full group $(t_2, r_2)$ .

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.