---
title: "Normal Subgroups and Quotient Groups"
description: "Cosets can be multiplied consistently exactly when the subgroup is normal. The quotient group that results turns the kernel of a homomorphism into a complete description of the map."
canonical: https://lightmysky.com/learn/mathematics/normal-subgroups-and-quotient-groups-mt_RU7A-t7wQ5
source: https://lightmysky.com/learn/mathematics/normal-subgroups-and-quotient-groups-mt_RU7A-t7wQ5.md
retrieved: 2026-09-12
---

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# Normal Subgroups and Quotient Groups

Cosets can be multiplied consistently exactly when the subgroup is normal. The quotient group that results turns the kernel of a homomorphism into a complete description of the map.

Subject: Mathematics · Area: Abstract Algebra · Ages 22 to 23
Page: https://lightmysky.com/learn/mathematics/normal-subgroups-and-quotient-groups-mt_RU7A-t7wQ5

## Ready when they can

- Decide whether a subgroup is normal, by conjugation or by its index
- Write the multiplication table of a small quotient group
- Use the first isomorphism theorem to identify the image of a homomorphism

## Lesson: When cosets form a group

You test normality by conjugation: a subgroup is normal when conjugating any of its elements by any group element stays inside it. You may skip the work in two cases, since every subgroup of an abelian group is normal and every subgroup of index 2 is normal. For a failure, conjugate the swap of 1 and 2 by the swap of 1 and 3 to get the swap of 2 and 3, which escapes.

**Example.** You find quotient sizes by dividing: a group of order 6 by a subgroup of order 2 leaves a quotient of order 3. The even residues modulo 6 are 0, 2, and 4, so that subgroup has 3 elements and yields 2 cosets, the evens and the odds. The symmetries of the square number 8 with a center of 2, so that quotient has order 4.

You read every homomorphism through its kernel with the first isomorphism theorem: the image looks exactly like the domain modulo the kernel. A map from the cyclic group of order 6 onto the cyclic group of order 3 has kernel of order 2, and its image is the whole codomain. The sign map on permutations of 3 letters hits both 1 and minus 1, so its image is that pair.

**Tip.** You should check normality before you multiply cosets, because the product is consistent exactly when left and right cosets agree. In an abelian group you may skip the test, since every subgroup there is normal. Never write a quotient table until the check passes.

**Recap.** Normality makes coset multiplication consistent, quotients divide orders, and every image is a quotient by its kernel.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [Equivalence Relations and Partitions](https://lightmysky.com/learn/mathematics/equivalence-relations-and-partitions-mt_wgEupYwEUF)
- [Homomorphisms, Cosets and Lagrange's Theorem](https://lightmysky.com/learn/mathematics/homomorphisms-cosets-and-lagranges-theorem-mt_yKiZ8sg7yN)

## Opens up

- [The Galois Correspondence](https://lightmysky.com/learn/mathematics/the-galois-correspondence-mt_30kyliQR_L)
- [Decomposing a Representation from Its Character](https://lightmysky.com/learn/mathematics/decomposing-a-representation-from-its-character-mt_ccEcWW9xAa)
- [Group Actions, Orbits and the Class Equation](https://lightmysky.com/learn/mathematics/group-actions-orbits-and-the-class-equation-mt_urut0TFTOR)
- [The Sylow Theorems and Groups of Small Order](https://lightmysky.com/learn/mathematics/the-sylow-theorems-and-groups-of-small-order-mt_vg2VYkLuz8)