Subgroups, Cosets, and Lagrange's Theorem

Given a group, the first question is what lives inside it. Subgroups, the cosets they carve out, and the divisibility constraint of Lagrange's theorem are the basic tools of finite group theory. This page also introduces normal subgroups and quotient groups, the constructions that make homomorphisms and the whole structure theory of finite groups possible. It builds directly on the group axioms.

Subgroups

A subset $H\subseteq G$ is a subgroup ($H\le G$) if it is itself a group under the inherited operation. In practice one checks the subgroup criterion: $H$ is a subgroup iff $H\ne \varnothing$ and $$a,b\in H\Rightarrow a{b}^{-1}\in H.$$ (This single condition packages closure, identity, and inverses.) Every group has the trivial subgroup $\{e\}$ and the improper subgroup $G$ itself; all others are proper, non-trivial.

The cyclic subgroup generated by $g$ is $⟨g⟩=\{g^n:n\in \mathbb{Z}\}$; its order equals $\mathrm{ord}(g)$. More generally $⟨S⟩$ is the smallest subgroup containing $S$ — the intersection of all subgroups that contain $S$.

Cosets

For $g\in G$ and $H\le G$:

• the left coset is $gH=\{gh:h\in H\}$;
• the right coset is $Hg=\{hg:h\in H\}$.

Key facts (proved from the axioms):

• Every element lies in a coset ($g\in gH$ since $e\in H$).
• Two left cosets are either identical or disjoint: $g_{1}H=g_{2}H$ iff $g_2^{-1}{g}_1\in H$. Hence the left cosets partition $G$.
• All cosets have the same size $|gH|=|H|$, because $h\mapsto gh$ is a bijection $H\to gH$.

The number of left cosets is the index $[G:H]$. (The number of right cosets is the same, via $gH\mapsto H{g}^{-1}$.)

Lagrange's theorem

Because the left cosets partition $G$ into $[G:H]$ pieces each of size $|H|$:

Lagrange's theorem. If $G$ is finite and $H\le G$, then $$|G|=[G:H]|H|.$$ In particular $|H|$ divides $|G|$.

Corollaries.

• $\mathrm{ord}(g)\mid |G|$ for every $g$ (take $H=⟨g⟩$).
• $g^{|G|}=e$ for all $g$ (raise to the index). Specialising to ${\mathbb{Z}}_n^{\times }$ recovers Euler's theorem $a^{\phi (n)}\equiv 1(\mathrm{mod}n)$, and to prime $n$, Fermat's little theorem.
• A group of prime order $p$ has no proper non-trivial subgroups, hence is cyclic and generated by any non-identity element: $G\cong {\mathbb{Z}}_p$.

Caution — the converse fails. A divisor $d\mid |G|$ need not be the order of any subgroup. The smallest counterexample is $A_4$ (order $12$), which has no subgroup of order $6$. Partial converses do hold: Cauchy's theorem (a subgroup of order $p$ exists for every prime $p\mid |G|$) and the Sylow theorems (prime-power divisors are realised).

Normal subgroups

The conjugate of $H$ by $g$ is $$gH{g}^{-1}\equiv \{gh{g}^{-1}:h\in H\}\le G,$$ a subgroup isomorphic to $H$. A subgroup is normal ($H◃G$) if it is stable under all conjugation: $$gH{g}^{-1}=H\text{for all }g\in G,$$ equivalently $gH=Hg$ for all $g$ (every left coset is a right coset). Normal subgroups are exactly the kernels of homomorphisms (see homomorphisms.md) and exactly the subgroups for which the quotient below is well-defined.

• In an abelian group every subgroup is normal.
• The centre $Z(G)=\{z:zg=gz\forall g\}$ is always normal, as is any subgroup of the centre.
• A subgroup of index $2$ is always normal (the two left cosets are $H$ and its complement, which must match the two right cosets).

Example: in $SU(2)$ the centre $\{\pm 1\}$ is normal, and $SU(2)/\{\pm 1\}\cong SO(3)$ — see examples-classical.md.

Quotient groups

When $H◃G$, the set of cosets $G/H=\{gH:g\in G\}$ becomes a group under $$(g_{1}H)(g_{2}H)=(g_1{g}_2)H.$$ Normality is exactly what makes this product well-defined — independent of the representatives $g_i$. Indeed if $g_{1}H=g_1^{\prime }H$ and $g_{2}H=g_2^{\prime }H$, normality lets one slide the ambiguity through and conclude $g_1{g}_{2}H=g_1^{\prime }{g}_2^{\prime }H$.

• $|G/H|=[G:H]=|G|/|H|$ (finite case).
• $G/\{e\}\cong G$ and $G/G\cong \{e\}$.
• $\mathbb{Z}/n\mathbb{Z}={\mathbb{Z}}_n$ is the archetype.

The quotient records "$G$ with $H$ collapsed to the identity". The precise sense in which quotients and homomorphic images coincide is the first isomorphism theorem in homomorphisms.md.

The correspondence (lattice) theorem

For $H◃G$ there is an inclusion-preserving bijection $$\{\text{subgroups of }G\text{ containing }H\}⟷\{\text{subgroups of }G/H\},K\mapsto K/H,$$ under which normal subgroups correspond to normal subgroups. This is the tool that lets one read off the subgroup structure of a quotient, and it underpins the composition series machinery.

References

• Dummit & Foote, Abstract Algebra, §3.1–3.3.
• Artin, Algebra, Ch. 2 §8–10.