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