--- title: "Homomorphisms, Cosets and Lagrange's Theorem" description: "Homomorphisms preserve a group's operation. Left cosets of a subgroup partition the group, as do right cosets separately. For a finite group, Lagrange's theorem says that each subgroup's order divides" 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: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Homomorphisms, Cosets and Lagrange's Theorem Homomorphisms preserve a group's operation. Left cosets of a subgroup partition the group, as do right cosets separately. For a finite group, Lagrange's theorem says that each subgroup's order divides the group's order. Kernels connect these maps with quotient groups. 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 group homomorphism f satisfies f(ab) = f(a)f(b), using each group's operation. It sends the source identity to the target identity. Its kernel is the set of inputs sent to that identity. For additive groups, write f(a + b) = f(a) + f(b). Test a proposed map by computing both sides. For example, powers preserve multiplication on nonzero reals because (ab)^k = a^k b^k. Z_m denotes residues modulo m under addition. If d divides m, reducing these residues modulo d defines a homomorphism to Z_d. Its kernel consists of multiples of d among 0 through m - 1. For Z18 to Z3, these are 0, 3, 6, 9, 12 and 15. More generally, f(a) = f(b) exactly when b inverse times a is in the kernel. Thus a trivial kernel makes f injective. *(drawing: List source residues and retain those sent to the target identity.)* For a subgroup H of G, a left coset is aH = {ah : h in H}; a right coset is Ha = {ha : h in H}. Left cosets partition G, and right cosets partition G separately. Do not mix the two sides in this claim. In additive groups, write a + H. Multiplication by a is a bijection from H to aH, so each left coset has H's size. The right-sided map h to ha is also a bijection. **Example.** In additive Z14, H = {0, 7}. The coset 1 + H is {1, 8}. Representatives 0 through 6 produce seven disjoint cosets covering all fourteen residues. To recover H from a + H, subtract a from every member. Representatives a and b give the same additive coset exactly when a - b belongs to H. Suppose c belongs to left cosets aH and bH. Write c = ah1 = bh2 with h1, h2 in H. Then b inverse times a = h2 h1 inverse belongs to H. Therefore a = bk for some k in H, so aH = bkH = bH. Here kH = H because H closes under products and inverses. Thus shared left cosets coincide. The corresponding argument applies separately to right cosets. *(drawing: The shared element produces a subgroup element relating the two representatives.)* For finite G, its left cosets are disjoint, cover G and each have |H| elements. If there are k cosets, |G| = k|H|. Hence |H| divides |G| and k = |G|/|H|. This is Lagrange's theorem. A size that does not divide |G| is impossible. Divisibility alone does not guarantee that a subgroup exists. A group of order 22 cannot have a subgroup of order 4. *(drawing: Count equal-sized cosets by dividing the finite group size by the subgroup size.)* **Recap.** Check that maps preserve the operation, partition into cosets, and use divisibility to rule out subgroup sizes. *Lesson and questions written for this dataset.* ## 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) ## Optional connections - [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)