Abelian group
Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.
The only bit that survives
A shuffle can be called even or odd, and the label behaves under composition. Ask whether some cleverer label — a number out of three, or out of four — could behave the same way, and the answer is that nothing else can — one bit is exactly what a permutation gives up.
Five-eighths of the pairs, and no more
Pick two symmetries of a square at random and do them in both orders: forty times in sixty-four the result is the same. No group that fails to commute does better. The reason is a count of pairs that turns into a count of conjugacy classes, and a two-line argument about the centre that caps the answer at five-eighths — reached by the square and the quaternions, approached from above by nothing, and approached from below by groups that commute a little more than half the time.
What homology forgets about a loop
Let the letters of a loop commute and the loop group of a space becomes its first homology group: a loop now records only how often it went round each hole. What is thrown away is exactly the loops that bound a surface. On the figure eight that is nearly everything — of the loops of sixty letters that homology calls nought, about one in 6,700 is a loop that actually shrinks.
Three coordinates and a fourth power
The Heisenberg group is made of triples of whole numbers, and two moves generate all of it. Count the triples within r moves of the start and the count grows like r⁴, not r³: at forty-four moves there are 1,600,703 of them, against 117,569 for the cubic lattice. The fourth power comes from one coordinate that the moves reach by enclosing area, so that height costs only the square root of itself — and seen from far away, the ball of reachable triples is a curved solid with a dimple at each pole.
The zero of a sandpile
Pile grains of sand on a grid, and whenever a cell holds four, let it topple and pass one grain to each neighbour. The order in which cells topple turns out not to matter, and the piles that keep recurring when sand is dropped at random form a group — one whose size is the number of spanning trees of the grid. Its zero is not the empty grid but a fractal picture of nested squares, and sand dropped on a single cell spreads into a round pile with the same kind of pattern inside.
Named alongside it
The objects these essays reach for when they reach for this one.
CommutatorConjugacy classCayley graphCellular automatonCommutativityCovering spaceDeterminantDihedral groupExhaustive searchFractalFree groupFundamental group