Subgroups, Cosets, and Lagrange's Theorem

  • A subgroup H ≤ G is a subset of group G that forms a group under the same operation.
  • Cosets partition a group into equal-sized pieces; left coset aH = {ah : h ∈ H}.
  • Lagrange's Theorem: |H| divides |G| for any subgroup H of finite group G.
  • Order of element divides group order; group of prime order p has only trivial subgroups.
  • Cosets are equivalence classes that establish a bijection between left and right cosets.
  • Emerged from 1770s permutation studies by Lagrange; formalized by Jordan and Weber.

Subgroups, Cosets, and Lagrange's Theorem

Overview

In the study of groups, one of the most fundamental questions is: what can we say about the structure of a group based on its substructures? Just as understanding the factors of an integer helps us understand the number itself, understanding the subgroups of a group—smaller groups contained within it—reveals deep information about the group's architecture. Cosets, which partition a group into equal-sized pieces relative to a subgroup, provide the key tool for this analysis. The crowning achievement of this elementary theory is Lagrange's Theorem, which establishes a surprising arithmetic constraint: the size of any subgroup must divide the size of the whole group. This theorem, simple to state yet profound in its consequences, is one of the first major results students encounter in abstract algebra. It connects the algebraic structure of groups to basic number theory and provides a powerful tool for both proving groups cannot exist with certain properties and for understanding those that do. Historically, these ideas emerged from the study of polynomial equations and permutations in the early 19th century. While Joseph-Louis Lagrange never stated the theorem in its modern group-theoretic form, his 1770 work on permutations contained the essential ideas. The formalization came later with the development of abstract group theory by mathematicians like Camille Jordan and later Heinrich Weber, who established groups as objects of study in their own right.

💡 The Core Insight

Subgroups partition the group into equal-sized cosets, and this partition reveals that the order of any subgroup must divide the order of the group—a purely algebraic structure imposes an arithmetic constraint.

Definition

Subgroup: Let (G,⋅)(G, \cdot) be a group. A subset H⊆GH \subseteq G is a subgroup of GG (written H≤GH \leq G) if HH forms a group under the same operation.

Left Coset: Given a subgroup H≤GH \leq G and an element g∈Gg \in G, the left coset of HH by gg is the set gH={gh:h∈H}gH = \{gh : h \in H\}.

Right Coset: The right coset of HH by gg is Hg={hg:h∈H}Hg = \{hg : h \in H\}.

Index: The index of HH in GG, denoted [G:H][G:H], is the number of distinct left (or right) cosets of HH in GG.

Lagrange's Theorem: If GG is a finite group and H≤GH \leq G, then ∣H∣|H| divides ∣G∣|G|, and specifically: |G| = |H| \cdot [G:H]

Core Theory

TheoremLagrange's Theorem

Theorem [1]: Let GG be a finite group and let HH be a subgroup of GG. Then: 1. The order of HH divides the order of GG 2. ∣G∣=∣H∣⋅[G:H]|G| = |H| \cdot [G:H] 3. For any g∈Gg \in G, the order of gg divides ∣G∣|G|

Proof

We prove this in several steps.

Step 1: All cosets of HH have the same size as HH. Let g∈Gg \in G and consider the map ϕg:H→gH\phi_g: H \to gH defined by ϕg(h)=gh\phi_g(h) = gh. To show ϕg\phi_g is injective: if ϕg(h1)=ϕg(h2)\phi_g(h_1) = \phi_g(h_2), then gh1=gh2gh_1 = gh_2, so by left cancellation (which holds in any group), h1=h2h_1 = h_2. To show ϕg\phi_g is surjective: any element of gHgH has the form ghgh for some h∈Hh \in H, so every element is in the image. Therefore ϕg\phi_g is a bijection, so ∣gH∣=∣H∣|gH| = |H|.

Step 2: The cosets of HH partition GG. First, every element belongs to some coset: for any g∈Gg \in G, we have g=g⋅e∈gHg = g \cdot e \in gH (where ee is the identity). Second, we show that two cosets are either identical or disjoint. Suppose g1H∩g2H≠∅g_1H \cap g_2H \neq \emptyset. Then there exist h1,h2∈Hh_1, h_2 \in H such that g1h1=g2h2g_1h_1 = g_2h_2. For any g1h∈g1Hg_1h \in g_1H, we have:

g1h=g1(h1h1−1)h=(g1h1)(h1−1h)=(g2h2)(h1−1h)=g2(h2h1−1h)g_1h = g_1(h_1h_1^{-1})h = (g_1h_1)(h_1^{-1}h) = (g_2h_2)(h_1^{-1}h) = g_2(h_2h_1^{-1}h)

Since HH is a subgroup, h2h1−1h∈Hh_2h_1^{-1}h \in H, so g1h∈g2Hg_1h \in g_2H. Thus g1H⊆g2Hg_1H \subseteq g_2H. By symmetry, g2H⊆g1Hg_2H \subseteq g_1H, so g1H=g2Hg_1H = g_2H.

Step 3: Computing the order. Let the distinct cosets be g1H,g2H,…,gkHg_1H, g_2H, \ldots, g_kH where k=[G:H]k = [G:H]. Since they partition GG and each has size ∣H∣|H|:

∣G∣=∑i=1k∣giH∣=∑i=1k∣H∣=k∣H∣=[G:H]⋅∣H∣|G| = \sum_{i=1}^{k} |g_iH| = \sum_{i=1}^{k} |H| = k|H| = [G:H] \cdot |H|

Therefore ∣H∣|H| divides ∣G∣|G|.

Step 4: Order of an element divides ∣G∣|G|. For any g∈Gg \in G, consider the cyclic subgroup ⟨g⟩={e,g,g2,…,gk−1}\langle g \rangle = \{e, g, g^2, \ldots, g^{k-1}\} where kk is the order of gg. Since ⟨g⟩≤G\langle g \rangle \leq G, by the above, ∣⟨g⟩∣|\langle g \rangle| divides ∣G∣|G|. But ∣⟨g⟩∣=k|\langle g \rangle| = k, so the order of gg divides ∣G∣|G|. □

∎
⚠️ Common mistake

Common Error: Lagrange's Theorem does not say that if dd divides ∣G∣|G|, then there exists a subgroup of order dd. The converse is false! For example, the alternating group A4A_4 has order 12, but there is no subgroup of order 6. The theorem only guarantees that subgroup orders divide the group order, not that all divisors correspond to subgroups.

mathematician

Joseph-Louis Lagrange (1770) — In his memoir on the algebraic solution of equations, Lagrange proved that the order of a permutation subgroup divides the order of the full symmetric group, establishing the principle that became Lagrange's Theorem.

visualization

Coset Partition of Z₆ by Subgroup H = {0, 3} 0 3 1 4 2 5 0 + H (Subgroup H) 1 + H 2 + H |H| = 2 |1+H| = 2 |2+H| = 2 |Z₆| = 6 |H| = 2 [Z₆:H] = 3

Examples and Applications

Example1

Subgroups of Z8\mathbb{Z}_8 under addition modulo 8 Consider the group Z8={0,1,2,3,4,5,6,7}\mathbb{Z}_8 = \{0, 1, 2, 3, 4, 5, 6, 7\} under addition modulo 8. This group has order 8, so by Lagrange's Theorem, any subgroup must have order dividing 8, i.e., order 1, 2, 4, or 8. Let's examine H={0,2,4,6}H = \{0, 2, 4, 6\}:

  • Identity: 0∈H0 \in H ✓
  • Closure: 2+2=4∈H2 + 2 = 4 \in H, 2+4=6∈H2 + 4 = 6 \in H, 4+4=0∈H4 + 4 = 0 \in H, etc. ✓
  • Inverses: −2≡6(mod8)-2 \equiv 6 \pmod{8}, and 6∈H6 \in H ✓

So H≤Z8H \leq \mathbb{Z}_8 with ∣H∣=4|H| = 4. The left cosets of HH are:

  • 0+H={0,2,4,6}=H0 + H = \{0, 2, 4, 6\} = H
  • 1+H={1,3,5,7}1 + H = \{1, 3, 5, 7\}
  • 2+H={2,4,6,0}=H2 + H = \{2, 4, 6, 0\} = H
  • 3+H={3,5,7,1}=1+H3 + H = \{3, 5, 7, 1\} = 1 + H

So there are exactly 2 distinct cosets, and [Z8:H]=2[\mathbb{Z}_8:H] = 2. Indeed, ∣H∣⋅[Z8:H]=4⋅2=8=∣Z8∣|H| \cdot [\mathbb{Z}_8:H] = 4 \cdot 2 = 8 = |\mathbb{Z}_8| ✓

Example2

Using Lagrange's Theorem to prove non-existence Can there exist a group of order 15 with an element of order 8? By Lagrange's Theorem, the order of any element must divide the order of the group. Since 8∤158 \nmid 15, no such element can exist. Therefore, any group of order 15 has elements only of orders 1, 3, 5, or 15. This example shows the power of Lagrange's Theorem for ruling out group structures without explicit construction.

Counterexample

Why we need HH to be a subgroup Consider G=S3G = S_3 (the symmetric group on 3 elements, with order 6) and let S={e,(12)}S = \{e, (12)\} where (12)(12) is a transposition. Computing S⋅SS \cdot S: we have (12)⋅(12)=e(12) \cdot (12) = e, so the products are {e,(12)}\{e, (12)\}. However, SS is not closed under composition because we'd need all products, and more importantly, (12)(13)=(132)∉S(12)(13) = (132) \notin S. If we compute "cosets" anyway:

  • e⋅S={e,(12)}e \cdot S = \{e, (12)\}, size 2
  • (13)⋅S={(13),(132)}(13) \cdot S = \{(13), (132)\}, size 2
  • (23)⋅S={(23),(123)}(23) \cdot S = \{(23), (123)\}, size 2

These sets partition GG and all have size 2, but SS is not a subgroup (it's not closed). This shows that while the coset construction is mechanical, Lagrange's Theorem's conclusion requires HH to actually be a subgroup—only then do we get the guarantee that coset sizes equal ∣H∣|H| and that they partition the group properly.

🧠 Intuition

Geometric Picture: Think of a group as a collection of symmetries, and a subgroup as a smaller collection of symmetries. The cosets represent "translated" copies of this smaller collection throughout the whole group. Lagrange's Theorem says that these translated copies tile the entire group perfectly with no overlap—like how you can tile a floor with identical tiles. For example, in the integers under addition, the subgroup 3Z={…,−6,−3,0,3,6,…}3\mathbb{Z} = \{\ldots, -6, -3, 0, 3, 6, \ldots\} represents "multiples of 3." The cosets are:

  • 0+3Z=3Z0 + 3\mathbb{Z} = 3\mathbb{Z} (numbers divisible by 3)
  • 1+3Z={…,−2,1,4,7,…}1 + 3\mathbb{Z} = \{\ldots, -2, 1, 4, 7, \ldots\} (numbers leaving remainder 1)
  • 2+3Z={…,−1,2,5,8,…}2 + 3\mathbb{Z} = \{\ldots, -1, 2, 5, 8, \ldots\} (numbers leaving remainder 2)

These three cosets partition all integers—every integer falls into exactly one category based on its remainder modulo 3.

Historical Development

The origins of Lagrange's Theorem lie in Joseph-Louis Lagrange's 1770 memoir Réflexions sur la résolution algébrique des équations, where he studied the permutations of roots of polynomial equations. Although Lagrange worked with permutations concretely rather than abstractly, he proved that what we would now call the index of a permutation subgroup equals the number of distinct values a rational function of the roots can take. The abstraction to general groups came much later. In 1870, Camille Jordan published his Traité des substitutions et des équations algébriques, which systematized the theory of permutation groups and stated results very close to the modern theorem. The final abstraction to arbitrary groups appeared in the late 19th century work of Walther von Dyck (1882), who defined groups axiomatically, and Heinrich Weber (1893), who developed abstract group theory systematically. The term "coset" (from "complex" and "set") was introduced in the early 20th century as group theory became established as an independent subject. The modern formulation and proof presented in textbooks today was standardized by Emmy Noether and her school in the 1920s as part of the broader movement toward structural mathematics.

Connections to Other Areas

connection

Connection to Quotient Structures and <a class="wiki-link" href="/wiki/modular-arithmetic">Modular Arithmetic</a>: The cosets of a subgroup form the foundation for quotient groups when the subgroup is normal. This construction directly generalizes modular arithmetic: the integers modulo nn, written Z/nZ\mathbb{Z}/n\mathbb{Z}, are precisely the cosets of nZn\mathbb{Z} in Z\mathbb{Z}. The notation Z/nZ\mathbb{Z}/n\mathbb{Z} literally means "the integers divided by (modulo) the subgroup nZn\mathbb{Z}." This connection unifies elementary number theory with abstract algebra and explains why congruence relations create well-defined arithmetic operations—they're secretly quotient groups in disguise.

What You Need First

What This Opens Up

See Also

Cyclic Groups · Quotient Groups · Normal Subgroups · Isomorphism Theorems · Sylow Theorems · Index of a Subgroup