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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
Four axioms, endless examples, and one test for subsets.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.