---
title: "Homomorphisms, Cosets and Lagrange's Theorem"
description: "Maps that preserve the operation, the classes a subgroup carves out, and the counting result that a subgroup's order divides the group's. Kernels are what make quotient groups possible."
canonical: https://lightmysky.com/learn/mathematics/homomorphisms-cosets-and-lagranges-theorem-mt_yKiZ8sg7yN
source: https://lightmysky.com/learn/mathematics/homomorphisms-cosets-and-lagranges-theorem-mt_yKiZ8sg7yN.md
retrieved: 2026-09-12
---

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# Homomorphisms, Cosets and Lagrange's Theorem

Maps that preserve the operation, the classes a subgroup carves out, and the counting result that a subgroup's order divides the group's. Kernels are what make quotient groups possible.

Subject: Mathematics · Area: Abstract Algebra · Ages 21 to 22
Page: https://lightmysky.com/learn/mathematics/homomorphisms-cosets-and-lagranges-theorem-mt_yKiZ8sg7yN

## Ready when they can

- Show a given map is a homomorphism and find its kernel
- Partition a small group into cosets of a subgroup
- Use Lagrange's theorem to rule out a subgroup of a stated size

## Lesson: Maps, slices, and counting

A homomorphism is a map between groups that preserves the operation: combining first and then mapping gives the same result as mapping first and then combining. The kernel is the set of inputs that land on the identity of the target, and it measures how much the map collapses. For the map sending each integer n to its remainder mod 5, the kernel is exactly the multiples of 5.

Cosets slice a group into equal non overlapping pieces. Adding a fixed element to every member of a subgroup gives one coset, and any two cosets are either exactly the same set or share nothing at all. In Z8 with subgroup H equal to 0 and 4, there are 8 divided by 2, which is 4 distinct cosets.

**Example.** Send each number in Z12 to its remainder mod 4. The outputs landing on 0 come from 0, 4, and 8, so the kernel has 3 elements. In Z6 with H equal to 0 and 3, the coset 2 plus H is 2 and 5, found by adding 2 to each member.

**Tip.** Lagrange theorem counts through cosets: the order of a subgroup divides the order of the group. Use it to rule sizes out. A group of order 7 is prime, so it has no proper nontrivial subgroup, since no number strictly between 1 and 7 divides 7.

**Recap.** Preserve the operation, slice into cosets, and let divisibility rule sizes out.

## Practice

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

## Needs first

- [Groups, Subgroups and Symmetry](https://lightmysky.com/learn/mathematics/groups-subgroups-and-symmetry-mt_HFAhcaGo2L)
- [Equivalence Relations and Partitions](https://lightmysky.com/learn/mathematics/equivalence-relations-and-partitions-mt_wgEupYwEUF)

## Opens up

- [Fermat's Little Theorem and Euler's Theorem](https://lightmysky.com/learn/mathematics/fermats-little-theorem-and-eulers-theorem-mt_7wW4z4Eidx)
- [Rings, Fields and Their First Properties](https://lightmysky.com/learn/mathematics/rings-fields-and-their-first-properties-mt_BmhIook13o)
- [Normal Subgroups and Quotient Groups](https://lightmysky.com/learn/mathematics/normal-subgroups-and-quotient-groups-mt_RU7A-t7wQ5)