Mathematics.

group theory

Lagrange's Theorem

Abstract Algebra I30 minDifficulty5 out of 10

You should know: subgroups

Overview

Lagrange's theorem states that for any finite group G and any subgroup H of G, the order of H divides the order of G, with the quotient |G|/|H| equal to the index [G:H], the number of distinct cosets of H. The proof is a clean counting argument: the cosets of H partition G into equal-sized blocks. Named after Joseph-Louis Lagrange, who proved a special case for permutation groups before the group concept was formalized, this theorem is one of the most heavily used facts in finite group theory — it instantly rules out many possible subgroup orders.

Intuition

The left cosets of H — sets of the form gH — partition G into non-overlapping blocks that are all exactly the same size as H (the map h ↦ gh is a bijection from H to gH). Since G is chopped up into some number of equal-size blocks, that number of blocks times the block size must equal |G|, forcing |H| to divide |G|. It's the same reasoning as noting that if you can perfectly tile a rectangle of area 12 with identical tiles, the tile's area must divide 12.

Formal Definition

Definition

Let G be a finite group and H ≤ G a subgroup. Then:

∣G∣=[G:H]⋅∣H∣|G| = [G:H] \cdot |H|
Lagrange's theorem
∣H∣ ∣ ∣G∣|H| \ \big|\ |G|
H's order divides G's order
[G:H]=number of distinct left cosets gH[G:H] = \text{number of distinct left cosets } gH
Index of H in G
∣a∣ ∣ ∣G∣for any a∈G|a| \ \big|\ |G| \quad \text{for any } a \in G
Corollary: element orders also divide |G|

Worked Examples

  1. By Lagrange's theorem, any subgroup order must divide 15.

    ∣H∣∣15|H| \mid 15

  2. The divisors of 15 = 3 × 5 are 1, 3, 5, 15.

    divisors(15)={1,3,5,15}\text{divisors}(15) = \{1,3,5,15\}

Answer: Possible subgroup orders are exactly 1, 3, 5, and 15.

Practice Problems

Difficulty 4/10

A group has order 20. Which of the following CANNOT be the order of a subgroup?

Difficulty 6/10

Show that a group G of prime order p has no nontrivial proper subgroups.

Difficulty 5/10

In a group of order 12, what must be true about the order of any single element a ∈ G?

Quiz

Lagrange's theorem states that for a finite group G and subgroup H:
A group of order 7 (prime) can have subgroups of which orders?
The converse of Lagrange's theorem (every divisor of |G| is the order of some subgroup) is:

Summary

  • For finite G and subgroup H ≤ G: |G| = [G:H]·|H|, so |H| always divides |G|.
  • Corollary: the order of every element of G divides |G|.
  • The converse is false in general — not every divisor of |G| need be a realized subgroup order (e.g. A₄ has no subgroup of order 6).

References

  1. WebsiteWikipedia — Lagrange's theorem (group theory)