Normal subgroups & quotient groups

Lagrange tells us how a subgroup slices G; but cosets generally don't form a group. Only when the subgroup is normal does the set of cosets inherit a group structure — giving the quotient group $G/N$. This is among the deepest ideas in abstract algebra.

20.1 Where do normal subgroups come from? Kernels of homomorphisms

Every group homomorphism $\phi :G\to H$ has a kernel$$\ker \phi :=\{g\in G:\phi (g)=e_H\}$$which is automatically normal in G. Conversely (First Isomorphism Theorem), every normal subgroup is the kernel of some homomorphism; moreover$$G/\ker \phi \cong \mathrm{im}\phi .$$

This ties §17's homomorphisms, §5's invariants, and §20's normal subgroups together: the three conservation laws are three homomorphisms G → small Abelian groups. Their kernels are three special normal subgroups of G:

• ${\Sigma }_{co}:G\to \mathbb{Z}/3$, kernel = states with corner twist sum 0
• ${\Sigma }_{eo}:G\to \mathbb{Z}/2$, kernel = states with edge flip sum 0
• $\mathrm{sgn}:G\to \mathbb{Z}/2$, kernel = states with even-permutation parity

20.2 Interactive: pick N, see G/N

20.3 Commutator subgroup [G,G] and abelianization

The commutator subgroup $[G,G]$ is generated by all commutators $[a,b]=ab{a}^{-1}{b}^{-1}$. It is always normal in G. The quotient $G^{\mathrm{ab}}:=G/[G,G]$ is called the abelianization — the "largest Abelian quotient" of G.

For the cube, it is known that $G^{\mathrm{ab}}\cong \mathbb{Z}/2$. The only Abelian shadow of G is the sgn homomorphism — half of G dies in [G,G]; only one bit (sign) survives. G is extremely non-Abelian.

20.4 Composition series & simple groups

Iteratively factoring G by normal subgroups gives a composition series$$G=G_0▹G_1▹G_2▹\cdots ▹G_n=\{e\}$$where each quotient $G_i/G_{i+1}$ is simple (no proper non-trivial normal subgroup). Jordan–Hölder guarantees that, up to reordering, this sequence is unique.

For the cube group, the simple factors are$$A_8\times A_{12}\times (\mathbb{Z}/2{)}^4\times (\mathbb{Z}/3{)}^7$$— essentially a restatement of the structure theorem from §6. A₈ and A₁₂ are members of an infinite family of non-Abelian finite simple groups.

20.5 Second & third isomorphism theorems

A clean cube application: take $H=⟨U,D⟩$ (states reachable using only U and D), and $N=[G,G]$ (states of even parity). Then $HN$ consists of all "⟨U,D⟩-states with arbitrary parity adjust," and by the second isomorphism theorem

The right-hand index [⟨U,D⟩ : ⟨U,D⟩ ∩ [G,G]] = 2, because each of U and D is itself an odd permutation. So HN partitions over [G,G] into exactly two cosets — a direct proof of the "⟨U,D⟩-based parity detector" used in many BLD methods.

Application on the cube: with $N=[G,G]$, $G/N\cong \mathbb{Z}/2$. ℤ/2 has only two subgroups ({e} and itself), so there are exactly two subgroups of G containing [G,G]: namely [G,G] and G. This precisely says "no intermediate subgroup sits strictly between [G,G] and G" — the sgn bit is indivisible.