Groups, Subgroups, and Cosets

⚡ Key Takeaways

Groups, Subgroups, and Cosets — A group is a set with an associative binary operation, an identity, and inverses. A subgroup is a subset that is itself a group under the same operation. Cosets partition a group into equal-sized translates of a subgroup, yielding Lagrange's Theorem: the order of a subgroup divides the order of the group. Applies throughout algebra, number theory, geometry, physics (symmetry), and cryptography. Foundational concepts formalized by Galois, Cayley, and others in the 19th century.

💡 The Core Insight

A group encodes "doing and undoing" — every action can be reversed. A subgroup is a self-contained world of such actions sitting inside a larger one. The miracle is that this smaller world tiles the bigger one perfectly: the cosets of $H$ are like identical floor tiles, all the same size, covering $G$ with no gaps and no overlaps. Counting tiles gives Lagrange's Theorem, and the way these tiles interact (when they form a group themselves) leads to quotient groups and the deepest structural theorems of algebra.

DefinitionGroup

A group is an ordered pair $(G, \cdot)$ where $G$ is a non-empty set and $\cdot : G \times G \to G$ is a binary operation satisfying:

1

\quad \forall a,b \in G : \quad a \cdot b \in G \quad \text{(closure)}

Derivation hint: This is built into the codomain of the operation $\cdot : G \times G \to G$ ; the output never escapes $G$ .

2

\quad \forall a,b,c \in G : \quad (a \cdot b) \cdot c = a \cdot (b \cdot c) \quad \text{(associativity)}

Derivation hint: Associativity lets us write $abc$ unambiguously; it is an axiom, not derived, and fails for operations like subtraction.

3

\quad \exists\, e \in G \ \ \forall a \in G : \quad e \cdot a = a \cdot e = a \quad \text{(identity)}

Derivation hint: $e$ acts neutrally on both sides; uniqueness follows since two identities $e, e'$ give $e = e \cdot e' = e'$ .

4

\quad \forall a \in G \ \ \exists\, a^{-1} \in G : \quad a \cdot a^{-1} = a^{-1} \cdot a = e \quad \text{(inverses)}

Derivation hint: Each element has a two-sided inverse; uniqueness follows from $b = b(ac) = (ba)c = c$ when $b,c$ are both inverses of $a$ . The symbols: $G$ is the underlying set, $\cdot$ the operation (often written multiplicatively as $ab$ or omitted), $e$ the identity, $a^{-1}$ the inverse of $a$ . The order of $G$ , denoted $|G|$ , is its cardinality. A group is abelian (commutative) if additionally $ab = ba$ for all $a,b$ .

DefinitionSubgroup

A subset $H \subseteq G$ is a subgroup of $G$ , written $H \leq G$ , if $H$ is non-empty and forms a group under the restriction of $\cdot$ . Equivalently (the one-step subgroup test):

5

\quad H \leq G \iff H \neq \varnothing \ \text{ and }\ \forall a,b \in H : \ a \cdot b^{-1} \in H

Derivation hint: Taking $a=b$ gives $e \in H$ ; then $a=e$ gives $b^{-1} \in H$ ; then closure under $\cdot$ follows from $a(b^{-1})^{-1} = ab \in H$ . The trivial subgroup is $\{e\}$ ; $G$ is a subgroup of itself. Any other is a proper, nontrivial subgroup. The subgroup generated by $a \in G$ is $\langle a \rangle = \{ a^n : n \in \mathbb{Z}\}$ , a cyclic subgroup.

DefinitionCosets and Index

For $H \leq G$ and $g \in G$ , the left coset of $H$ by $g$ and the right coset are

6

\quad gH = \{\, gh : h \in H \,\}, \qquad Hg = \{\, hg : h \in H \,\}

Derivation hint: A coset is the image of $H$ under left- (or right-) translation by $g$ ; it is generally not a subgroup unless $g \in H$ . The set of all distinct left cosets is denoted $G/H$ (read " $G$ mod $H$ "). The index of $H$ in $G$ is the number of distinct left cosets:

7

\quad [G : H] = |G/H| = \#\{\text{distinct left cosets } gH\}

Derivation hint: Each coset has the same cardinality $|H|$ via the bijection $h \mapsto gh$ , so the cosets are equal-sized blocks.

TheoremLagrange's Theorem

Let $G$ be a finite group and $H \leq G$ . Then $|H|$ divides $|G|$ , and

8

\quad |G| = [G : H]\cdot |H|

9

\quad a^{|G|} = e \quad \text{for all } a \in G

Derivation hint: Let $n = \mathrm{ord}(a)$ . By Lagrange $n \mid |G|$ , say $|G| = nk$ ; then $a^{|G|} = (a^n)^k = e^k = e$ .

ProofCosets Partition the Group; Lagrange Follows

We prove that the left cosets of $H \leq G$ form a partition of $G$ , all of equal size, and deduce Lagrange's Theorem.

Step 1 — Every element lies in some coset. For any $g \in G$ , since $e \in H$ we have $g = ge \in gH$ . Thus $\bigcup_{g \in G} gH = G$ (the cosets cover $G$ ).

Step 2 — Two cosets are equal or disjoint. Suppose $g_1 H \cap g_2 H \neq \varnothing$ ; pick $x$ in the intersection. Then $x = g_1 h_1 = g_2 h_2$ for some $h_1, h_2 \in H$ , giving

g_1 = g_2 h_2 h_1^{-1}.

Since $H$ is a subgroup, $h_2 h_1^{-1} \in H$ . For any element $g_1 h \in g_1 H$ ,

g_1 h = g_2 (h_2 h_1^{-1} h) \in g_2 H,

so $g_1 H \subseteq g_2 H$ . By symmetry $g_2 H \subseteq g_1 H$ , hence $g_1 H = g_2 H$ . Therefore distinct cosets are disjoint.

Step 3 — All cosets have the same size. The map $\varphi : H \to gH$ defined by $\varphi(h) = gh$ is a bijection: it is surjective by definition of $gH$ , and injective because $gh_1 = gh_2 \implies h_1 = h_2$ (left-multiply by $g^{-1}$ ). Hence $|gH| = |H|$ for every $g$ .

Step 4 — Count. By Steps 1–2, $G$ is the disjoint union of exactly $[G:H]$ left cosets; by Step 3 each has $|H|$ elements. Therefore

|G| = [G:H]\cdot|H|,

which is equation [8]. Since $[G:H]$ is a positive integer, $|H| \mid |G|$ . $\blacksquare$

∎

⚠️ Common mistake

Lagrange's Theorem is a one-way statement: it says $|H| \mid |G|$ , but it does not guarantee that for every divisor $d$ of $|G|$ there exists a subgroup of order $d$ . The converse fails (see the counterexample below). Partial converses exist — Cauchy's Theorem (a subgroup of prime order $p$ exists whenever $p \mid |G|$ ) and the Sylow Theorems for prime-power divisors.

mathematician · Joseph-Louis Lagrange | 1736–1813

Lagrange, an Italian-French mathematician of immense breadth, never worked with the abstract group concept — it did not exist in his lifetime. In his 1770–71 study of the solvability of polynomial equations (Réflexions sur la résolution algébrique des équations), he examined how the roots of an equation are permuted and how many distinct values a rational function of the roots can take. The number of such values, he observed, divides $n!$ — the seed of what we now call "Lagrange's Theorem." The general group-theoretic statement was crystallized only decades later by Galois, Cauchy, and Jordan. Lagrange's name attached to the theorem honors this originating insight even though the modern framing is anachronistic.

🧠 Intuition

Think of a group as the full set of moves you can make on some symmetric object — say all the ways to reposition a square so it lands on itself (the dihedral group). A subgroup is a self-consistent subset of those moves: for instance, "only rotations." Now translate the rotation subgroup by a single reflection: you get the coset of all rotations-followed-by-that-reflection. There are exactly two cosets here — pure rotations and rotation+reflection — each containing the same number of moves. The group splits into a clean stack of identical layers. The deep point: a subgroup imposes a grid on the group. Picking $gH$ is like asking "which layer am I in?" Lagrange's count is just "number of layers $\times$ layer thickness = total." This is why $|H|$ must divide $|G|$ — you cannot tile a floor with identical tiles unless the tile area divides the floor area.

ExampleThe integers mod 12 and a subgroup

Given: $G = (\mathbb{Z}_{12}, +)$ , the integers $\{0,1,\dots,11\}$ under addition mod 12, with $|G| = 12$ . Let $H = \langle 4 \rangle = \{0, 4, 8\}$ .

Step 1 — Verify $H$ is a subgroup. Check $H$ is closed and contains inverses: $4+8 = 12 \equiv 0$ , $4+4 = 8$ , $8+8 = 16 \equiv 4$ , inverse of $4$ is $8$ , inverse of $8$ is $4$ , inverse of $0$ is $0$ . So $H \leq G$ with $|H| = 3$ .

Step 2 — Form the (left = right, since abelian) cosets. Using additive notation $g + H$ :

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

Step 3 — Count cosets. There are $4$ distinct cosets, so $[G:H] = 4$ .

Step 4 — Verify Lagrange. $|G| = [G:H]\cdot|H| = 4 \times 3 = 12.$ ✓

\boxed{\,[G:H] = 4,\qquad |H| = 3,\qquad 4 \cdot 3 = 12 = |G|\,}

(All quantities dimensionless — counts of elements.)

ExampleCosets of rotations inside the dihedral group

D_4

Given: $G = D_4$ , the symmetry group of the square, with $|G| = 8$ . Elements: rotations $\{e, r, r^2, r^3\}$ (by $0°,90°,180°,270°$ ) and reflections $\{s, sr, sr^2, sr^3\}$ . Let $H = \langle r \rangle = \{e, r, r^2, r^3\}$ , the rotation subgroup, $|H| = 4$ .

Step 1 — Left cosets. Compute $gH$ for representatives:

$eH = \{e, r, r^2, r^3\}$ (all rotations)
$sH = \{s, sr, sr^2, sr^3\}$ (all reflections)

Step 2 — Are there others? Any rotation $r^k$ gives $r^k H = H$ . Any reflection $g$ gives $gH = sH$ . So exactly two cosets.

Step 3 — Index and Lagrange. $[G:H] = 2$ , hence $|G| = 2 \times 4 = 8$ . ✓

Step 4 — Bonus observation. Because the index is $2$ , $H$ is automatically a normal subgroup (left cosets $=$ right cosets), and the quotient $G/H \cong \mathbb{Z}_2$ records "rotation vs. reflection" parity.

\boxed{\,[D_4 : \langle r\rangle] = 2,\qquad |\langle r\rangle| = 4,\qquad 2\cdot 4 = 8 = |D_4|\,}

⚠️ Common mistake

Cosets are usually not subgroups. The coset $gH$ contains the identity $e$ if and only if $g \in H$ , in which case $gH = H$ itself. For $g \notin H$ , the coset $gH$ has no identity element and is therefore not a group. Treating $gH$ as a subgroup is a frequent and fatal error.

Left cosets ≠ right cosets in general. $gH$ and $Hg$ coincide for all $g$ precisely when $H$ is normal. In a non-abelian group with a non-normal subgroup, the left and right partitions differ, even though both have the same number of blocks $[G:H]$ .

CounterexampleThe converse of Lagrange fails

Lagrange says every subgroup's order divides $|G|$ . The converse — "for every divisor $d$ of $|G|$ there is a subgroup of order $d$ " — is false. Take $G = A_4$ , the alternating group on $4$ symbols, with $|A_4| = 12$ . The divisor $d = 6$ divides $12$ , yet $A_4$ has no subgroup of order $6$ . Why: A subgroup $H$ of order $6$ would have index $2$ , hence be normal, and $G/H \cong \mathbb{Z}_2$ . Then for every $g$ , $g^2 \in H$ . But $A_4$ contains the eight $3$ -cycles, each of order $3$ ; for a $3$ -cycle $\sigma$ , $\sigma = \sigma^4 = (\sigma^2)^2 \in H$ , so all eight $3$ -cycles would lie in $H$ . Together with $e$ that already exceeds $6$ elements — contradiction. Hence no subgroup of order $6$ exists.

connection

Cosets are the engine behind the most important constructions in algebra. When a subgroup is normal, its cosets themselves form a group under $(aH)(bH) = (ab)H$ — the quotient group $G/H$ — which feeds directly into the Isomorphism Theorems and the study of Normal Subgroups and Quotient Groups. Lagrange's element-order corollary [9], applied to $G = (\mathbb{Z}/n\mathbb{Z})^\times$ , yields Euler's Theorem and Fermat's Little Theorem in number theory. The index and coset machinery generalizes to Group Actions and the Orbit-Stabilizer Theorem, where orbits play the role of cosets of stabilizers. Cyclic subgroups $\langle a \rangle$ connect to Cyclic Groups and Generators, and the whole apparatus extends to Sylow Theorems (partial converses of Lagrange).

🔬 Where this appears

⏳ History

- 1770–1771 — Lagrange studies permutations of roots of polynomials and observes that the number of values of a symmetric function divides

n!

— the proto-form of Lagrange's Theorem.

- 1799 — Ruffini attempts to prove the unsolvability of the quintic, implicitly using permutation-group ideas and subgroup structure.

- 1815 — Cauchy publishes systematic work on permutations and "substitutions," introducing cycle notation and proving early structural results.

- 1830–1832 — Évariste Galois introduces the term groupe, normal subgroups, and the connection between solvability of equations and group structure — the conceptual birth of group theory. He dies in a duel at 20; his work is published posthumously in 1846.

- 1854 — Arthur Cayley gives the first abstract definition of a finite group via multiplication tables ("On the theory of groups, as depending on the symbolic equation

\theta^n = 1

").

- 1870 — Camille Jordan publishes Traité des substitutions, the first comprehensive treatise, consolidating Galois theory and proving deep results about permutation groups.

- 1872 — Ludwig Sylow proves the Sylow Theorems, the most powerful partial converses to Lagrange's Theorem.

- 1882 — Walther von Dyck gives the modern abstract axiomatic definition of a group, including infinite groups, freeing the concept from permutations entirely.

- 1897 — William Burnside publishes Theory of Groups of Finite Order, the first English-language treatise, shaping 20th-century finite group theory.

open-problem

While the basic theory is complete, several deep threads radiate from these foundations. The converse-of-Lagrange question — which divisors of $|G|$ are actually orders of subgroups — has no simple answer; the full picture requires the Sylow theorems, Hall subgroups (which exist for all "divisor-coprime" orders only in solvable groups, a theorem of Philip Hall), and detailed structure theory. The Classification of Finite Simple Groups (completed ~2004, spanning ~10,000+ journal pages) is the monumental endpoint of the program begun with Lagrange and Galois: it lists the "atoms" from which all finite groups are built via quotients and extensions. Open frontiers include the study of coset geometries and buildings (Tits), efficient algorithms for the coset enumeration problem (Todd–Coxeter, undecidable in full generality for infinite presentations), and quantitative questions about subgroup growth in infinite groups.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also