Subgroups, Cosets, and Lagrange's Theorem

A subgroup $H$ of a group $(G,\cdot )$ is a nonempty subset closed under the group operation and inverses, so that $(H,\cdot )$ is a group in its own right. Given $H$, the left cosets $gH=\{gh:h\in H\}$ form a partition of $G$ into blocks all of the same size $|H|$, and this single combinatorial fact yields Lagrange's Theorem: $|H|$ divides $|G|$ for finite $G$. The result matters because it is the first powerful constraint on the possible sizes of subgroups and element orders, immediately giving Fermat's Little Theorem, the fact that groups of prime order are cyclic, and a backbone for the entire structure theory of finite groups. The surprising fact: although $|H|$ must divide $|G|$, the converse fails — a divisor of $|G|$ need not be the order of any subgroup, the smallest counterexample being $A_4$, which has order 12 but no subgroup of order 6.

The Core Insight

A subgroup acts on its parent group by translation, and translation is a bijection. Therefore every coset has exactly as many elements as the subgroup, and the cosets tile the group without overlap. Counting tiles times tile-size gives the group's order — so the subgroup's order must divide it. Almost everything about finite groups starts from this counting argument.

Formal Statement

Subgroup

Let $(G,\cdot )$ be a group with identity $e$. A subset $H\subseteq G$ is a subgroup, written $H\le G$, if:

Equivalently (the one-step subgroup test), a nonempty subset $H$ is a subgroup iff:

Cosets and Index

Let $H\le G$ and $g\in G$. The left coset of $H$ by $g$ and the right coset are:

The index $[G:H]$ is the number of distinct left cosets of $H$ in $G$:

Lagrange's Theorem

If $G$ is a finite group and $H\le G$, then

In particular, for every $g\in G$ the order of $g$ (the least $n>0$ with $g^n=e$) divides $|G|$, since $⟨g⟩\le G$ has order equal to $\mathrm{ord}(g)$:

Why It Holds / Derivation

Cosets partition $G$ into equal-sized blocks

We prove Lagrange's Theorem by establishing two lemmas, then counting.

• reflexive: $a^{-1}a=e\in H$;
• symmetric: if $a^{-1}b\in H$ then $(a^{-1}b{)}^{-1}=b^{-1}a\in H$;
• transitive: if $a^{-1}b,b^{-1}c\in H$ then $(a^{-1}b)(b^{-1}c)=a^{-1}c\in H$.

Since the classes of an equivalence relation partition the set, the left cosets partition $G$: distinct cosets are disjoint and their union is all of $G$.

• surjective by definition of $gH$;
• injective because $g{h}_1=g{h}_2\Rightarrow h_1=h_2$ by left-cancellation.

Therefore $|H|$ divides $|G|$. $■$

Left vs. right cosets need not coincide

In general $gH\ne Hg$. They do coincide for every $g$ exactly when $H$ is a normal subgroup. But Lagrange's Theorem holds regardless: the number of left cosets equals the number of right cosets, because $gH\mapsto H{g}^{-1}$ is a bijection between the two sets of cosets, so $[G:H]$ is well-defined either way.

Joseph-Louis Lagrange | 1736–1813

Lagrange, working in pre-group-theoretic language, studied how the value of a polynomial in $n$ variables changes when its variables are permuted. In his 1770–71 memoir Réflexions sur la résolution algébrique des équations, he observed that the number of distinct values a rational function takes under all $n!$ permutations always divides $n!$. This is precisely the index formula $[S_n:H]$ applied to the stabilizer $H$ of the function — the embryonic form of "Lagrange's Theorem," decades before Galois, Cauchy, and Cayley gave groups their abstract definition. The theorem now bearing his name is the abstract distillation of this combinatorial observation.

Building Intuition

Think of a subgroup $H$ as a "pattern" or "tile" sitting inside $G$, anchored at the identity. Multiplying everything in $H$ by a fixed element $g$ slides the whole tile to a new location $gH$ without distorting it — translation is rigid, so the slid tile has the same number of points. Now slide the tile to every possible position. Two slid tiles either land on exactly the same spot (if $g_1^{-1}{g}_2\in H$) or are completely disjoint — there is no partial overlap, because membership in a coset is an all-or-nothing equivalence. So the group floor is tiled perfectly by congruent copies of $H$. Counting the floor area two ways (total points vs. tiles × points-per-tile) forces $|H|$ to divide $|G|$. The integers $\mathrm{mod}n$ make this concrete: $H=\{0,3\}\subseteq {\mathbb{Z}}_6$ tiles ${\mathbb{Z}}_6$ as $\{0,3\},\{1,4\},\{2,5\}$ — three tiles of two, and $3\times 2=6$.

Cosets of $H=\{0,3\}$ tiling $\mathbb{Z}_6$ into three equal blocks

Worked Examples

Cosets and index in $\mathbb{Z}_{12}$

• $0+H=\{0,4,8\}$
• $1+H=\{1,5,9\}$
• $2+H=\{2,6,10\}$
• $3+H=\{3,7,11\}$
• $4+H=\{4,8,0\}=0+H$ (repeat)

Element orders via Lagrange in $S_3$

• identity $e$: order 1.
• transpositions $(12),(13),(23)$: order 2 (each squares to $e$).
• 3-cycles $(123),(132)$: order 3.

Common Misconceptions

The converse of Lagrange is FALSE

Lagrange's Theorem says: if $H$ is a subgroup then $|H|\mid |G|$. It does not say that every divisor of $|G|$ is the order of some subgroup. Students routinely assume "divisor $\Rightarrow$ subgroup exists," which is wrong in general. Partial converses do hold:

• Cauchy's Theorem: if a prime $p$ divides $|G|$, there is an element (and subgroup) of order $p$.
• Sylow's Theorems: if $p^k$ is the largest power of $p$ dividing $|G|$, a subgroup of order $p^k$ exists.
• Finite abelian / cyclic groups: the full converse holds — every divisor is achieved.

$A_4$ has no subgroup of order 6

The alternating group $A_4$ has order $12$, and $6\mid 12$, yet $A_4$ has no subgroup of order 6. Suppose $H\le A_4$ with $|H|=6$. Then $[A_4:H]=2$, which would force $H$ to be normal, and the quotient $A_4/H\cong {\mathbb{Z}}_2$. Counting: $A_4$ contains eight 3-cycles, each of order 3. A subgroup of index 2 must contain the square of every element (since in $A_4/H\cong {\mathbb{Z}}_2$, $g^2\in H$ always); but for a 3-cycle $g$, $g=g^4=(g^2{)}^2\in H$, so all eight 3-cycles lie in $H$ — already more than 6 elements. Contradiction. Hence no order-6 subgroup exists, definitively refuting the converse of Lagrange.

Connections

Lagrange's Theorem is the gateway to the structural theory of finite groups. It directly feeds Cyclic Groups (a group of prime order is forced to be cyclic since every non-identity element has order $p$) and motivates Normal Subgroups and Quotient Groups, where cosets of a normal subgroup acquire their own group structure $G/N$. The proof's machinery — translating a subgroup by group elements — is a special case of Group Actions and the Orbit-Stabilizer Theorem, in which $|G|=|\text{orbit}|\cdot |\text{stabilizer}|$ generalizes [5]. Its partial converses are Cauchy's Theorem and Sylow Theorems. In number theory the corollary $g^{|G|}=e$ specializes, via the unit group $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$, to Euler's Theorem and Fermat's Little Theorem. The index also underlies the First Isomorphism Theorem through $|G/\ker \phi |=|\mathrm{im}\phi |$.

Real-World Applications

1. RSA cryptography. Euler's Theorem $a^{\phi (n)}\equiv 1(\mathrm{mod}n)$ — a Lagrange corollary in $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$ — underlies the correctness of RSA encryption/decryption with exponents inverse mod $\phi (n)$.
2. Primality testing. Fermat-based tests (and Miller–Rabin) use $a^{p-1}\equiv 1(\mathrm{mod}p)$, i.e. Lagrange in $(\mathbb{Z}/p\mathbb{Z}{)}^{\times }$ of order $p-1$.
3. Error-correcting codes. Cyclic codes are ideals in ${\mathbb{F}}_q[x]/(x^n-1)$; coset leaders and standard-array decoding partition the space into cosets of the code (a subgroup of ${\mathbb{F}}_q^n$).
4. Crystallography. The order of a crystal's point group must divide the order of its space-group operations restricted to a unit cell; coset decompositions classify symmetry-equivalent atomic positions (Wyckoff positions).
5. Rubik's Cube and puzzle theory. Reachable configurations form a subgroup of all $\sim 4.3\times 10^{19}$ states; cosets identify which scrambles are solvable, and the legal-move group's order divides the full state count.
6. Quantum mechanics / spectroscopy. Degeneracies of energy levels equal dimensions of irreducible representations whose count and sizes obey order/index constraints from the molecular symmetry group.
7. Music theory. Transpositions and inversions of the 12-tone system form ${\mathbb{Z}}_{12}$ and the dihedral group $D_{12}$; cosets classify pitch-class sets up to symmetry.

Historical Development

• 1770–1771 — Lagrange's memoir on equation-solving notes that the number of values of a polynomial under variable permutations divides $n!$ — the seed of the theorem.
• 1799 — Paolo Ruffini uses permutation-substitution counting (implicitly coset-like) in attempts to prove the unsolvability of the quintic.
• 1815 — Augustin-Louis Cauchy systematically studies "substitutions" (permutations), proving results on cycle structure and laying groundwork for subgroup counting.
• 1830–1832 — Évariste Galois introduces normal subgroups ("invariant subgroups") and proper decompositions, deepening the coset idea in his theory of solvability.
• 1845–1846 — Cauchy proves what is now Cauchy's Theorem (prime divisors yield elements of that order), a partial converse to Lagrange.
• 1854 — Arthur Cayley gives the first abstract definition of a group in On the theory of groups, as depending on the symbolic equation ${\theta }^n=1$, allowing Lagrange's result to be stated for abstract groups.
• 1872 — Ludwig Sylow publishes the Sylow Theorems, the deepest partial converses, guaranteeing subgroups of prime-power order.
• 1870s–1880s — Camille Jordan and Walther von Dyck consolidate abstract group theory; cosets, index, and Lagrange's Theorem take their modern textbook form.
• 1897 — William Burnside's Theory of Groups of Finite Order presents the topic in essentially the form taught today.

Subtleties & Limits

• Counting subgroups of given order. Determining the number of subgroups of each order in an arbitrary finite group is computationally and theoretically intricate; for $p$-groups it connects to deep questions about subgroup lattices and Hall's enumeration formulas.
• CLT groups. A group satisfying the full converse of Lagrange (a subgroup of every order dividing $|G|$) is called a converse-Lagrange-theorem (CLT) group. Characterizing CLT groups is subtle: all supersolvable groups are CLT, but the property is not closed under quotients or subgroups in obvious ways, and a clean intrinsic characterization remains messy.
• Infinite groups. For infinite $G$ the equation $|G|=[G:H]|H|$ becomes a statement about cardinal arithmetic and generally gives no useful constraint; "order divides order" loses meaning, so Lagrange's combinatorial force is genuinely a finiteness phenomenon.
• Beyond groups. Analogues for loops, quasigroups, and Moufang loops are delicate — Lagrange's Theorem can fail for general quasigroups, and its validity for Moufang loops was a long-studied question resolved only with substantial machinery.

See Also