Stabiliser
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
Twenty-four ways to set a cube down
Count the rotations of a cube from its corners and the answer is eight times three. Count from its edges and it is twelve times two; from its faces, six times four. Three different pictures give one number because each count is the same theorem — the places a thing can go, times the motions that leave it where it is — and the same theorem splits Cayley's sixteen trees into twelve and four and proves that a group of eight has a centre.
Necklaces made of symmetries
Lagrange's theorem says a subgroup's size divides the group's, and the converse is false. One piece of the converse is true: every prime that divides the size is the order of some element. The proof threads the group's own elements onto a necklace whose product is nothing, turns it, and counts — the argument that proved Fermat's little theorem with beads, with the beads replaced by motions.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Dihedral groupConjugacy classCounting argumentGroup actionLagrange theoremOrbitAbelian groupAlternating groupAutomorphismCommutativityCommutatorCoset