---
title: "Groups, Subgroups and Symmetry"
description: "One operation, associative, with an identity and inverses. Symmetries of an object, permutations and modular arithmetic all satisfy the same four axioms, so anything proved once holds for all of them."
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: 2026-09-12
---

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# Groups, Subgroups and Symmetry

One operation, associative, with an identity and inverses. Symmetries of an object, permutations and modular arithmetic all satisfy the same four axioms, so anything proved once holds for all of them.

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 one operation that meets four axioms. The operation must stay inside the set, combine in any grouping, provide an identity element that changes nothing, and give every element an inverse that undoes it. Anything proved from those four rules alone holds for every group, from numbers to symmetries to permutations.

**Example.** The integers under addition form a group: sums stay integers, grouping never matters, zero is the identity, and every number has its negative as inverse. Positive integers under addition fail because there is no identity and no inverses inside the set. Integers under subtraction fail too, since grouping changes the answer.

The symmetry group of a square collects every motion that leaves the square looking unchanged. There are eight: four rotations including doing nothing, and four reflections across the two diagonals plus the vertical and horizontal midlines. Listing all eight element by element is the fastest way to feel the axioms at work.

**Tip.** To test a candidate subset, use the subgroup test: it must contain the identity and stay closed under the operation and inverses. Watch the empty set trap, since the empty set can never contain the identity. Practise the test on subsets of small groups before arguing abstractly.

**Recap.** Four axioms, endless examples, and one test for subsets.

## Practice

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

## Needs first

- [Linear Maps and Their Matrices](https://lightmysky.com/learn/mathematics/linear-maps-and-their-matrices-mt_pzrfiQFdto)
- [Equivalence Relations and Partitions](https://lightmysky.com/learn/mathematics/equivalence-relations-and-partitions-mt_wgEupYwEUF)

## 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)