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 Group Theory|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.

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.

Core Theory

Lagrange's Theorem

1. The order of $H$ divides the order of $G$
2. $|G|=|H|\cdot [G:H]$
3. For any $g\in G$, the order of $g$ divides $|G|$

We prove this in several steps.

Let $g\in G$ and consider the map ${\varphi }_g:H\to gH$ defined by ${\varphi }_g(h)=gh$.

To show ${\varphi }_g$ is injective: if ${\varphi }_g(h_1)={\varphi }_g(h_2)$, then $g{h}_1=g{h}_2$, so by left cancellation (which holds in any group), $h_1=h_2$.

To show ${\varphi }_g$ is surjective: any element of $gH$ has the form $gh$ for some $h\in H$, so every element is in the image.

Therefore ${\varphi }_g$ is a Bijection|bijection, so $|gH|=|H|$.

First, every element belongs to some coset: for any $g\in G$, we have $g=g\cdot e\in gH$ (where $e$ is the identity).

Second, we show that two cosets are either identical or disjoint. Suppose $g_{1}H\cap g_{2}H\ne \varnothing$. Then there exist $h_1,h_2\in H$ such that $g_1{h}_1=g_2{h}_2$. For any $g_{1}h\in g_{1}H$, we have:

Since $H$ is a subgroup, $h_2{h}_1^{-1}h\in H$, so $g_{1}h\in g_{2}H$. Thus $g_{1}H\subseteq g_{2}H$. By symmetry, $g_{2}H\subseteq g_{1}H$, so $g_{1}H=g_{2}H$.

Let the distinct cosets be $g_{1}H,g_{2}H,\dots ,g_{k}H$ where $k=[G:H]$. Since they partition $G$ and each has size $|H|$:

Therefore $|H|$ divides $|G|$.

For any $g\in G$, consider the Cyclic Group|cyclic subgroup $⟨g⟩=\{e,g,g^2,\dots ,g^{k-1}\}$ where $k$ is the order of $g$. Since $⟨g⟩\le G$, by the above, $|⟨g⟩|$ divides $|G|$. But $|⟨g⟩|=k$, so the order of $g$ divides $|G|$. □

Examples and Applications

1

Consider the group ${\mathbb{Z}}_8=\{0,1,2,3,4,5,6,7\}$ 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 $H=\{0,2,4,6\}$:

• Identity: $0\in H$ ✓
• Closure: $2+2=4\in H$, $2+4=6\in H$, $4+4=0\in H$, etc. ✓
• Inverses: $-2\equiv 6(\mathrm{mod}8)$, and $6\in H$ ✓

So $H\le {\mathbb{Z}}_8$ with $|H|=4$.

The left cosets of $H$ are:

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

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

2

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 $8\nmid 15$, 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.

Consider $G=S_3$ (the Symmetric Group|symmetric group on 3 elements, with order 6) and let $S=\{e,(12)\}$ where $(12)$ is a transposition.

Computing $S\cdot S$: we have $(12)\cdot (12)=e$, so the products are $\{e,(12)\}$. However, $S$ is not closed under composition because we'd need all products, and more importantly, $(12)(13)=(132)\notin S$.

If we compute "cosets" anyway:

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

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

For example, in the integers under addition, the subgroup $3\mathbb{Z}=\{\dots ,-6,-3,0,3,6,\dots \}$ represents "multiples of 3." The cosets are:

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

What You Need First

What This Opens Up

See Also