--- title: "Groups, Subgroups and Symmetry" description: "A group is a set with a closed, associative operation, an identity and inverses. Integers under addition, permutations under composition and square symmetries share these rules. The choice of set and " canonical: https://lightmysky.com/learn/mathematics/groups-subgroups-and-symmetry-mt_HFAhcaGo2L source: https://lightmysky.com/learn/mathematics/groups-subgroups-and-symmetry-mt_HFAhcaGo2L.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`. # Groups, Subgroups and Symmetry A group is a set with a closed, associative operation, an identity and inverses. Integers under addition, permutations under composition and square symmetries share these rules. The choice of set and operation matters. Subject: Mathematics · Area: Abstract Algebra · Ages 21 to 22 Page: https://lightmysky.com/learn/mathematics/groups-subgroups-and-symmetry-mt_HFAhcaGo2L ## Ready when they can - Check the group axioms for a given set and operation - Describe the symmetry group of a square and list its elements - Use the subgroup test on a candidate subset ## Lesson: One rule for many symmetries A group is a set with an operation satisfying four axioms. Closure keeps every result in the set. Associativity means (a * b) * c = a * (b * c). An identity e satisfies e * a = a * e = a. Each a has an inverse b with a * b = b * a = e. The symbol * stands for the chosen operation. These rules do not require a * b = b * a. **Example.** Integers under addition form a group. Sums stay integers, addition is associative, 0 is the identity, and the inverse of n is -n. Positive integers exclude 0 and additive inverses, so they fail. Integer subtraction is closed but not associative: (9 - 4) - 2 = 3, while 9 - (4 - 2) = 7. One failed axiom is enough to rule out a group. *(drawing: The same numbers give different results when subtraction is regrouped.)* Modulo 6, use the congruence classes represented by 0, 1, 2, 3, 4 and 5. Add or multiply, then reduce to a remainder. Addition is closed and associative; reduction preserves the remainder of either grouping. Its identity is 0. The additive inverse of a is the remainder of -a, so every class has one. Multiplication has identity 1 but fails the inverse axiom: 0 times any class is 0, never 1. **Example.** A multiplicative inverse modulo n must give remainder 1 when multiplied by the original element. A smaller subset can pass even when all classes fail. Modulo 4, the subset {1, 3} is a group under multiplication: 1 times 1 gives 1, 1 times 3 and 3 times 1 give 3, and 3 times 3 gives remainder 1. The identity is 1, each element is its own inverse, and associativity is inherited. *(drawing: For addition, an element and its inverse give the identity remainder 0.)* A square's planar symmetry group contains the distance-preserving motions that map it onto itself. The operation is composition: perform one motion, then another. Its eight elements are rotations about the center through 0, 90, 180 and 270 degrees, and reflections across the vertical midline, horizontal midline and two diagonals. Composing symmetries stays inside the set and is associative. Doing nothing is the identity; reversing a motion gives its inverse. *(drawing: Two half-turns give the identity motion.)* A symmetry can preserve the outline while moving vertices. Its order is the least positive number of repetitions that returns every vertex to its own starting position. A half-turn has order 2. A clockwise turn through t degrees, with 0 < t < 360, is undone by a clockwise turn through 360 - t degrees. A reflection undoes itself. A subgroup is a subset of a group using the same operation. Test that it contains the parent's identity and is closed under the operation and inverses. Associativity is inherited. Another test asks for a nonempty subset closed under the operation and inverses: any element combined with its inverse then supplies the identity. The empty set passes those closure statements vacuously but contains no identity, so it is not a subgroup. **Example.** For multiples of a fixed integer k under addition, 0 = k times 0, ka + kb = k(a + b), and -(ka) = k(-a). These prove identity, closure and inverses for the whole subset. In a modular group, reduce each result before checking membership. For a finite candidate, check every pair for closure; one successful pair does not prove it. **Recap.** Four axioms, endless examples, and one test for subsets. *The lesson revisions, questions and worked solutions were written for this dataset.* ## Practice 14 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) ## Optional connections - [Linear Maps and Their Matrices](https://lightmysky.com/learn/mathematics/linear-maps-and-their-matrices-mt_pzrfiQFdto) ## Opens up - [Noether's Theorem: Symmetry and Conserved Quantities](https://lightmysky.com/learn/science/noethers-theorem-symmetry-and-conserved-quantities-mt_kAoOn8syBK) - [Primitive Roots and the Units Modulo n](https://lightmysky.com/learn/mathematics/primitive-roots-and-the-units-modulo-n-mt_KwTDckkMze) - [Group Actions, Orbits and the Class Equation](https://lightmysky.com/learn/mathematics/group-actions-orbits-and-the-class-equation-mt_urut0TFTOR) - [Homomorphisms, Cosets and Lagrange's Theorem](https://lightmysky.com/learn/mathematics/homomorphisms-cosets-and-lagranges-theorem-mt_yKiZ8sg7yN) - [The Fundamental Group and Loops on a Circle](https://lightmysky.com/learn/mathematics/the-fundamental-group-and-loops-on-a-circle-mt_zH50_L0nu4)