Concept

Coset — where it appears

One of the equal-sized shifted copies a subgroup cuts the whole group into. Their equal size is what forces a subgroup's order to divide the group's, which is Lagrange's theorem in one line.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

The sixteen words of the [7,4] Hamming code. A table of sixteen seven-bit codewords with their data bits, parity bits and weights.

Sixteen spheres that fill a cube

A hundred and twenty-eight seven-bit words, sixteen of them chosen, and a ball of eight around each. Sixteen times eight is a hundred and twenty-eight exactly — so the balls tile the space with nothing left over, and the code wastes nothing at all.

computation · Error-correcting codes

The syndrome of 1011010, and the bit it names. A parity-check matrix over a received word, with the resulting syndrome matched against the table of single-error syndromes.

Finding the error without reading the message

Three parity checks on a seven-bit word produce three bits. If they are all zero nothing is wrong; otherwise they are the number of the position that broke. The message is never consulted, because the answer does not depend on it.

computation · Error-correcting codes

A subgroup of 2, and the 4 blocks it cuts the group into. The 8 symmetries of a 4-gon, split into 4 blocks by composing every element onto the subgroup {e, r²}. The blocks all have 2 elements and no element is in two of them.

The blocks a subgroup cuts out

Take any part of a group that is closed under composition, and it slices the whole group into blocks of its own size that do not overlap. Everything Lagrange's theorem says is arithmetic about that picture — and whether the blocks can be multiplied is a separate question with a surprising answer.

algebra · Symmetry groups

The group drawn as a map

A multiplication table says everything about a group and shows nothing; lay the same information out as one dot per element and one arrow per generator, and multiplying becomes walking, distance becomes a word length, and the group acquires a shape.

algebra · Cayley graph

Orbit times stabiliser is 8, on every row. A table of 5 things the 8 symmetries of a 4-gon can move. Each row draws every position the thing can be carried to and lists the motions that leave it where it is; the two counts multiply to 8 on every row.

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.

algebra · Symmetry groups

The 3 periods of the 13th roots of unity, and the equation they solve. Roots of unity modulo 13 grouped by the cosets of the index-3 subgroup; the periods 0.274, 1.377, −2.651 are the roots of x³ + x² − 4x + 1.

The sums that obey a smaller equation

The twelve non-trivial thirteenth roots of unity satisfy an equation of degree twelve. Split them into three groups of four — the right three groups — and add each group: the three sums are the roots of x³ + x² − 4x + 1, an equation of degree three with whole-number coefficients. Gauss called such sums periods, found one for every divisor of p − 1, and used them to build the seventeen-gon from four quadratic equations.

algebra · Roots of unity

Named alongside it

The objects these essays reach for when they reach for this one.

Counting argument Dihedral group Group action Cyclic group Error-correcting code Hamming code Lagrange theorem Automorphism Cayley graph Conjugacy class Constructible number Cyclotomic polynomial

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