Normal Subgroups and Quotient Groups

Cosets chopped a group G into equal-sized, non-overlapping tiles: the left cosets gH of a subgroup H. That felt tantalisingly close to something bigger — if the tiles are all the same size and perfectly partition the group, could we treat each whole tile as a single new "super-element" and build a smaller group out of them?

Sometimes yes, sometimes no. The tiles form a group exactly when the subgroup is normal — the one extra condition that makes coset multiplication behave. When it holds, the resulting group of tiles is called the quotient group (or factor group) G/N, and it is one of the most powerful constructions in all of algebra: it is how we "divide" one group by another, how we collapse detail we don't care about, and — via homomorphisms — how every structure-preserving map secretly encodes a quotient.

This one page is about that single idea: what makes a subgroup normal, and why exactly that condition is the key that unlocks the quotient group.

The definition of a normal subgroup

A subgroup N \le G is normal, written N \triangleleft G, if any one of these equivalent conditions holds for every g \in G:

The slogan to memorise is "left cosets equal right cosets." Ordinarily gH and Hg are different sets — normality is precisely the demand that they coincide, so there is only one family of cosets to talk about, not two competing ones.

Why normality is exactly what makes the cosets a group

We want to multiply cosets by the natural rule

(aN)(bN) = (ab)N.

This looks obvious, but there is a hidden danger: a coset can be named many different ways. The coset aN is the same set as a'N whenever a' = an for some n \in N. For the rule to define a genuine operation on tiles, the answer (ab)N must not depend on which representatives a, b we happened to grab. This is called being well-defined, and it is the whole ball game.

Take another pair of names a' = an_1 and b' = bn_2 with n_1, n_2 \in N. Then

a'b' = a n_1 \, b\, n_2 = a b\,(b^{-1} n_1 b)\, n_2.

For this to land in the same coset (ab)N, we need the middle piece b^{-1} n_1 b to sit inside N — that is exactly b^{-1} N b \subseteq N, the normality condition! If N is normal, b^{-1} n_1 b \in N, so a'b' \in abN and the product is unambiguous. If N is not normal, you can find representatives that give genuinely different answers, and the "operation" isn't an operation at all.

See it: collapsing \mathbb{Z}_6 onto \mathbb{Z}_3

Let G = \mathbb{Z}_6 = \{0,1,2,3,4,5\} under addition mod 6, and take N = \langle 3 \rangle = \{0, 3\}. Because \mathbb{Z}_6 is abelian, N is automatically normal. Its cosets are the three tiles

\{0,3\}, \qquad 1+N = \{1,4\}, \qquad 2+N = \{2,5\}.

The figure below reveals the partition, then shows how adding tiles — e.g. (1+N) + (2+N) = 3+N = N — makes the three cosets behave exactly like \mathbb{Z}_3 = \{0,1,2\}. Step through it and watch the six-element group collapse into a three-element one.

This is the archetype of the whole subject: \mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}_n. The infinite group \mathbb{Z} modulo the subgroup n\mathbb{Z} of multiples of n is precisely "clock arithmetic" with n hours — the quotient is the mod- n world.

More worked quotients

Even and odd: S_n / A_n. The alternating group A_n (the even permutations) sits inside the symmetric group S_n with index 2. Any subgroup of index 2 is automatically normal (each of its two left cosets, and each of its two right cosets, is either the subgroup itself or "everything else"). So A_n \triangleleft S_n, and the quotient has just two elements:

S_n / A_n = \{\, A_n,\ \tau A_n \,\} \cong \{+1, -1\} \cong \mathbb{Z}_2,

where \tau is any single transposition. The two cosets are exactly "even" and "odd," and multiplying them follows the sign rule (-1)(-1) = +1 — odd composed with odd is even.

The centre is always normal. The centre Z(G) = \{ z : zg = gz \text{ for all } g\} is normal, because if z commutes with everything then gzg^{-1} = z g g^{-1} = z \in Z(G). So G / Z(G) always makes sense — a quotient that, remarkably, controls whether G itself is abelian.

Everything in an abelian group is normal. If G is abelian then gN = Ng trivially for every subgroup, so every subgroup is normal and every subgroup gives a quotient. Non-abelian groups are where normality becomes a real, discriminating condition.

Kernels: where normal subgroups come from

There is a beautiful two-way street between normal subgroups and homomorphisms.

Why is a kernel normal? For n \in \ker\varphi and any g,

\varphi(gng^{-1}) = \varphi(g)\,\varphi(n)\,\varphi(g)^{-1} = \varphi(g)\, e_H\, \varphi(g)^{-1} = e_H,

so gng^{-1} \in \ker\varphi — the kernel survives conjugation, which is exactly normality. Together these two facts say: normal subgroups are precisely the subgroups you can quotient by, and equivalently precisely the subgroups that arise as kernels. This is the doorway to the First Isomorphism Theorem, G/\ker\varphi \cong \operatorname{im}\varphi.

Trap 1 — "setwise" is not "elementwise." The condition gNg^{-1} = N is an equality of sets. It does not say gng^{-1} = n for each individual n (that stronger condition would make every element of N commute with g — that's being central, not merely normal). Normality only asks that conjugation permutes the elements of N among themselves; it may shuffle them around, as long as nothing leaves the set.

Trap 2 — normality is not transitive. It is tempting to think N \triangleleft H and H \triangleleft G force N \triangleleft G. They do not! The smallest counterexample lives in the symmetries of a square, D_4: you can have N \triangleleft H \triangleleft D_4 with N not normal in D_4, because an element of the big group can conjugate N out of place even though elements of H cannot. Normality is a relationship between a subgroup and one specific parent group — never assume it transfers up the chain.

A group with no normal subgroups except the two trivial ones (\{e\} and the whole group) is called simple. Simple groups are the atoms of group theory: because you can't quotient a simple group down into anything smaller, they play the role that prime numbers play in arithmetic. Just as every whole number factors into primes, every finite group can be broken apart, layer by layer via quotients, into a stack of simple pieces (its composition factors) — a result called the Jordan–Hölder theorem.

That analogy drove one of the great collective achievements of twentieth-century mathematics: the Classification of Finite Simple Groups, a proof spanning tens of thousands of pages by hundreds of authors. It says the finite simple groups fall into a few infinite families (cyclic groups of prime order, the alternating groups A_n for n \ge 5, and the groups of Lie type) plus exactly 26 sporadic exceptions — the largest of which, the "Monster," has more elements than there are atoms in Jupiter. And the humble alternating group A_5, of order 60, is the very smallest non-abelian simple group — the reason there's no formula for the roots of a general quintic.