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Expert ∞ Differential Equations, Abstract Algebra & Applied Math Topic 71 of 97

Cosets and Lagrange’s Theorem

The order of a subgroup divides the order of the group.

Key idea

The order of a subgroup divides the order of the group.

In depth

Cosets gH partition the group into equal-size pieces. Hence |H| divides |G| (Lagrange), and the order of any element divides |G|. The index [G:H] = |G|/|H|.

Worked example

A group of order 12 cannot have a subgroup of order 5.

Formula

|G| = [G:H]|H|

Real-world uses

Common mistake

Assuming the converse: a group of order 12 need not have a subgroup of order 6 (A₄ does not).

How to learn this

  1. Read the key idea and say it in your own words.
  2. Work through the worked example, then try a similar problem.
  3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

What is Cosets and Lagrange’s Theorem?

The order of a subgroup divides the order of the group. It is a topic at the expert level in Differential Equations, Abstract Algebra & Applied Math.

What is the formula for Cosets and Lagrange’s Theorem?

The key formula is |G| = [G:H]|H|.

Where is Cosets and Lagrange’s Theorem used in real life?

Real-world examples: Explaining why a group of 12 elements cannot have a subgroup of size 5.

What should I learn before Cosets and Lagrange’s Theorem?

Make sure you are comfortable with Dihedral Groups, the previous topic in Differential Equations, Abstract Algebra & Applied Math.

What is a common mistake with Cosets and Lagrange’s Theorem?

Assuming the converse: a group of order 12 need not have a subgroup of order 6 (A₄ does not).

How can I practise Cosets and Lagrange’s Theorem?

Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.

What comes after Cosets and Lagrange’s Theorem?

Move on to Normal Subgroups and Quotients next.

Related expert topics in Differential Equations, Abstract Algebra & Applied Math