Engineering / Mathematics — Abelian Groups
Cosets and Lagrange's Theorem
Cosets as a partition of a group, Lagrange's theorem, and its consequences for element orders.
Executive summary
The cosets of a subgroup partition the group into equal-sized blocks. Counting the blocks gives Lagrange's theorem, and Lagrange's theorem gives Fermat's little theorem and Euler's theorem as immediate corollaries.
It is the most consequential counting argument in elementary group theory.
Learning objectives
- Define cosets and prove they partition the group.
- State and prove Lagrange's theorem.
- Derive Fermat and Euler as corollaries.
01Cosets
Coset
For a subgroup H ≤ G and a ∈ G, the coset is aH = {ah : h ∈ H}.
In an abelian group left and right cosets coincide, so the distinction does not arise.
Cosets partition the group
Every element lies in exactly one coset of H, and all cosets have the same size |H|.
Reason. The relation a ~ b iff a⁻¹b ∈ H is an equivalence relation whose classes are the cosets. Each coset is in bijection with H via h ↦ ah, which is injective by cancellation.
Two facts do the work: the cosets are equivalence classes, so they partition; and left multiplication is a bijection, so they are equal in size. Neither requires commutativity.
02Lagrange's theorem
Lagrange's theorem
For a finite group G and subgroup H, |H| divides |G|, and the number of cosets is |G| / |H|, called the index of H.
The proof is immediate from the partition: |G| is the number of cosets times the common coset size.
Cosets partition G
Every element lies in exactly one.
All cosets have size |H|
Left multiplication is a bijection.
Count
|G| = (number of cosets) × |H|.
Conclude
|H| divides |G|.
03Corollaries
Applying Lagrange to the subgroup generated by an element gives the divisibility of element orders, from which the classical theorems follow directly.
| Statement | Derivation |
|---|---|
| ord(a) divides |G| | Apply Lagrange to ⟨a⟩, which has order ord(a) |
| a^{|G|} = e | Since ord(a) divides |G| |
| Euler: a^{φ(n)} ≡ 1 mod n | Take G = Z_n*, of order φ(n) |
| Fermat: a^{p−1} ≡ 1 mod p | Take G = Z_p*, of order p − 1 |
| Groups of prime order are cyclic | Any non-identity element generates a subgroup of order dividing p, hence order p |
04Frequently asked questions
Does the converse of Lagrange hold?
Not for general groups — a divisor of the group order need not correspond to a subgroup. For finite abelian groups the converse does hold, so every divisor gives a subgroup, and for cyclic groups that subgroup is unique.
Why does the theorem need finiteness?
The counting argument requires finite cardinalities. For infinite groups the index may still be defined and finite, and a version of the statement survives, but the divisibility formulation does not apply.
Is the index always an integer?
Yes, by the theorem itself — it is the number of cosets, which is a count. The content of Lagrange is precisely that this count times |H| recovers |G| exactly.
Sources and method
Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 190-194.
This page carries the durable method layer only: definitions, constructions, algorithms, complexity results and selection criteria, authored originally for KEVOS. No text is transcribed or paraphrased from the source, and no numeric tables or benchmark data are reproduced — these are routed to live authoritative sources instead.
Author: Kevin Jogin. Last reviewed 2026-08-07.