Abstract Algebra: Symmetry as a Structure of Its Own

Rotate a square tile by a quarter turn, then flip it over. Flip it first, then rotate it. Do you get the same result? Try it on a real object and you will find you do not. The order of the moves matters, and yet the full list of ways to put a square back into its own outline is a tidy, closed family: eight moves, each one undoable, any two combining into another on the list.

That family has a name, a group, and the course called abstract algebra starts from a daring idea. Do not study squares, clocks or cubes. Study the thing they share, the rules of combining, and prove everything that follows from the rules. Whatever you prove about groups is then true of every square, every clock and every cube at once.

A graduate does not remember the theorems so much as the habit: when something has symmetry, ask for its group, and count.

A group is a set of moves you can always undo

A group is a set of things with a way of combining any two into a third, obeying four rules:

• Closed: combining two members gives a member.
• Associative: (a then b) then c equals a then (b then c).
• Identity: some member e does nothing.
• Inverses: each member has a partner that undoes it.

Commutativity is not a rule. A group in which order never matters is called abelian; most interesting groups are not. A first proof, to feel the style: the identity is unique. Suppose e and e′ both do nothing. Then e combined with e′ equals e (since e′ does nothing) and equals e′ (since e does nothing). So e = e′. The trick is feeding each hypothesis to the other.

The smallest non-commutative group is the six ways of moving an equilateral triangle onto itself: three rotations (turn 0, 120, 240 degrees) and three flips. Call a 120-degree turn r and a flip s. "rs" will mean "do r, then s". Pick any two moves and see the result, and check whether the other order gives the same triangle.

The table at the bottom of the card is the group's Cayley table. Each element appears exactly once in every row and column, because you can always undo. A whole group, symmetries and all, is nothing but that table.

Clock groups and the idea of order

The simplest groups are clocks. Add hours on an n-hour clock and you have the group ℤ_n. Pick a step size a and keep adding it: how many steps until you come back to 0? That count is the order of a, and the places you visit form a subgroup, a group inside the group.

An element that visits everything is a generator; 5 generates ℤ₁₂ but 4 does not. This also settles when two groups are really the same. Two groups are isomorphic if you can relabel one into the other so the tables match. ℤ₆ is isomorphic to ℤ₂ × ℤ₃ (the 1 of ℤ₆ has order 6, and so does the pair (1, 1)). But ℤ₄ is not isomorphic to ℤ₂ × ℤ₂: the first has an element of order 4 and the second has none. Comparing element orders is the standard way to tell groups apart.

Lagrange's theorem: subgroups tile the group

Take a subgroup H inside a group G and "translate" it: pick any member g and form gH, everything you get by doing g and then a member of H. These translates are called cosets. Two facts make a picture. First, each coset has exactly as many members as H, because the move "do g first" can be undone, so it matches H one to one with gH. Second, any two cosets are either identical or share nothing. If they shared a member, you could rewrite one as the other. So the cosets cut G into equal-sized, non-overlapping tiles.

Count the tiles: |G| = (number of cosets) × |H|. That is Lagrange's theorem: the size of a subgroup always divides the size of the group. The tricks are that a translation is reversible, so it preserves size, and that two overlapping cosets must coincide. It was Lagrange (about 1771, in his study of polynomial equations) who noticed the pattern before groups existed as a concept.

The converse is false, and that is what makes the theorem a tool rather than a tautology. A group of order 12 need not have a subgroup of every size that divides 12. The alternating group on four things, which has 12 members, has no subgroup of order 6. Choose a group below and the page finds every subgroup by brute force.

A consequence you will use for life: in the multiplication of non-zero residues mod a prime p (which has p − 1 members), every element's order divides p − 1. Hence a^p−1 ≡ 1 (mod p), Fermat's little theorem, and the engine inside RSA encryption (see 12/09 and 34/75).

Invariants: what a move can never change

A homomorphism is a map between groups that respects the combining: do two moves and then translate equals translating each and then combining. It forgets detail but keeps structure, and its great use is catching things that cannot happen. The standard example is parity. Every shuffle of n things is a product of swaps of two items. A shuffle may be written with many swaps, but always an even number or always an odd number, never both. Call these even and odd, and the map "even → +1, odd → −1" respects combining (two odds make an even, as −1 × −1 = +1).

Now the 15 puzzle: a grid of sliding tiles with one gap. Every legal slide leaves the parity of a certain count (for the 3×3 version, the number of "inversions", pairs of tiles that are out of order) unchanged. Swap two tiles by lifting them out and you flip that parity. Those positions can never be reached by sliding, and exactly half of all arrangements are in that unreachable half. An 1879 paper by Johnson and Story proved this for the 15 puzzle. The trick is to find a quantity that no move changes, then show the target has a different value.

Press a tile next to the gap to slide it. Scramble as hard as you like: the inversion count stays even. The swap button produces a board that looks innocent and cannot be solved.

Rings and fields: which clocks let you divide

A group has one operation. A ring has two, addition and multiplication, linked by the distributive law: the integers, the clock numbers, polynomials and matrices are rings. A field is a ring where you can also divide by anything non-zero: the rationals, the reals, and the complex numbers.

Clock arithmetic is a ring for any n. Is it a field? Division by a means finding a number b with a·b ≡ 1. Such a b exists exactly when gcd(a, n) = 1, and Euclid's algorithm finds it (stepping back through the remainders). If gcd(a, n) > 1 you are in trouble: mod 12, 3 × 4 = 12 ≡ 0, so two non-zero numbers multiply to zero, and "zero divisors" make division impossible. So ℤ_n is a field exactly when n is prime. The trick behind inverses: if gcd = 1, Euclid's remainders can be run backwards to write 1 as a combination of a and n.

Finite fields are not a curiosity. They are the arithmetic behind error-correcting codes (QR codes, CDs, deep-space probes), elliptic-curve cryptography, and the AES cipher in your phone.

Rubik's cube, the unsolvable quintic, and the monster

The moves of a Rubik's cube form a group, and counting it uses everything above. Naively a cube has 8 corner pieces in 8 places with 3 twists each, and 12 edge pieces in 12 places with 2 flips each. Multiply: 8! × 3⁸ × 12! × 2¹². But real turns obey three rules (an invariant, again): the corner twists must total a multiple of 3, the edge flips must total an even number, and corners and edges must have the same permutation parity. Those cut the count by 3, 2 and 2, a factor of 12, leaving

A team led by Tomas Rokicki showed in 2010, using a lot of donated computer time, that every one of these positions can be solved in at most 20 face turns ("God's number" is 20). The 12 unreachable arrangements for every reachable one are the cube you get by prying it apart and reassembling it badly.

The same subject settles an old problem. The quadratic formula has cousins for cubics and quartics (16th century). For the general fifth-degree equation no formula using only +, −, ×, ÷ and roots exists. Abel and Ruffini gave proofs in the 1820s; Évariste Galois explained why a few years later, shortly before his death in 1832 at age 20. His idea: attach to each equation the group of ways to shuffle its roots while keeping all their algebraic relations. An equation is solvable by roots exactly when that group can be built up in simple abelian stages. The group for a general quintic is the full set of 120 shuffles of five things, which cannot be so built. The obstruction is a group-theoretic fact, with no pencil-and-paper cleverness needed.

That raises a question: what are the "atoms" of groups? A group with no inner structure to peel away is simple, and every finite group is built from simple ones. The classification of finite simple groups lists them all: a few infinite families (the cyclic groups of prime order, the alternating groups on five or more things, and several families of matrix groups) plus exactly 26 "sporadic" groups that fit no family. The largest, the Monster, has about 8 × 10⁵³ members. The classification is one of the largest proofs ever assembled, spread over tens of thousands of journal pages by many authors. Its completion was announced in 1983, a gap in one case was filled in 2004, and a streamlined "second-generation" proof (the Gorenstein–Lyons–Solomon programme) is still unfinished; the result is accepted by the field.

What a graduate just sees

• If something has symmetry, ask for its group and count it. The count often explains the object.
• The order of moves matters in most groups; "abelian" is the special, easy case.
• A subgroup's size always divides the group's size. If a number does not divide, no such subgroup exists.
• When a puzzle seems impossible, look for an invariant, a quantity that no legal move changes.
• Two structures are the same if their tables match after relabelling; element orders tell them apart.
• Clock arithmetic lets you divide exactly when the clock size is prime.
• Whether an equation can be solved by a formula is a statement about the group of its roots.

Where it shows up next

• Number Theory (34/75): the units mod n form a group, and Fermat's and Euler's theorems are Lagrange's theorem in disguise.
• Cryptography and coding: RSA uses the group of units mod a product of two primes; elliptic-curve systems use a group built from points on a curve; error-correcting codes use finite fields.
• Physics: conserved quantities come from symmetry (Noether's theorem), and the particle zoo is sorted by representations of symmetry groups. See Quantum Mechanics (34/83) and the Standard Model piece (13/07).
• Chemistry and materials: the 230 space groups classify every crystal; molecule vibrations are sorted by their symmetry group.
• Topology (34/76): algebraic topology attaches groups to shapes to tell them apart, the same "invariant" trick as the puzzle.

If you want to go deeper

• 📘 Textbook. Joseph Gallian, Contemporary Abstract Algebra (10th edition). The usual first text: many examples, gentle on proofs.
• 📘 Textbook. David Dummit and Richard Foote, Abstract Algebra (Wiley, 3rd edition). The comprehensive reference that mathematics students keep for years.
• 📘 Gentler. Nathan Carter, Visual Group Theory (MAA). Groups taught through pictures and Cayley diagrams, with the Rubik's cube in view; or Mark Ronan, Symmetry and the Monster (Oxford, 2006) for the story of the classification.
• 🌐 Free course. MIT OpenCourseWare 18.701 Algebra I (Fall 2010, taught by Michael Artin): the full undergraduate group, ring and field course with notes and problem sets.