Matpic
mathematics, in pictures
Concept

Subgroup

A subset of a group that is a group in its own right under the same operation, closed under combining and under inverses. A group's structure is read off its subgroups, and for a finite group each of their sizes divides its own.

Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.

A wedge of 2 circles. Several circles all passing through one common point, each labelled with a generator, so that a loop is a word in those letters.

The subgroup that is freer than the group

A free group on two letters contains a subgroup of index three that is free on four. Nothing about a group makes that plausible; everything about a graph makes it obvious, and the argument is to stop looking at the group and start looking at the space whose loops it is.

topology · Covering spaces The subgroups of a polynomial's symmetries, against the fields they name. Two lattices side by side, one of the subgroups of the symmetry group of the cube roots of two and the other of the fields between the rationals and the splitting field, drawn so that one is the other turned upside down.

The lattice that runs the other way

The symmetries of a polynomial's roots form a group, and the fields between the bottom and the top form a lattice. The two are the same picture, one of them turned over — a bigger group of symmetries fixes less, so it names a smaller field.

algebra · Galois correspondence 3 sheets, and the subgroup they name. A circle with its 3-sheeted cover drawn as a spiral above it, beside a table of the winding classes and whether each lifts to a closed loop. The ones that do are exactly the multiples of 3.

Every cover is a subgroup

A space can be unrolled, and the ways of unrolling it are not arbitrary. They correspond exactly to the subgroups of its fundamental group — index equals sheets, normality equals symmetry — so a question about a group becomes a question about a picture and back again.

topology · Homotopy 3 symmetries over 3 sheets: a regular covering. A 3-sheeted covering of a wedge of 2 circles, with the permutations of its sheets that commute with every generator. There are 3, against 3 sheets.

The symmetries a cover has of its own

A covering space can be shuffled without disturbing anything below it, and how many ways there are is decided by the subgroup it corresponds to. When there are as many symmetries as sheets the covering is called regular, and that is the same statement as the subgroup being normal.

topology · Covering spaces 3 sheets, 8 of 26 words coming back. A table of reduced words in two generators with the sheet each sends the base sheet to. The words returning to it are the covering's subgroup, and the 3 sheets are its cosets.

A covering is a permutation

Describing a covering means saying where each loop sends each sheet, which is a permutation for every generator. So a covering of a wedge of circles is nothing but a homomorphism to a symmetric group, and the subgroup it corresponds to is a stabiliser.

topology · Covering spaces 6 vertices folded to 4, and a graph that decides. The graph built from 3 generator words, folded until no vertex has two edges of one label leaving it. Reading a word from the base vertex decides membership, and 6 words are tested.

Folding a graph until it decides

A subgroup of a free group usually arrives as a list of words, and almost nothing about it is readable from the list. Draw the words as loops, merge every pair of edges with the same label leaving one point, and what is left is a machine that decides membership by reading.

topology · Covering spaces 3 sheets over a surface of genus 2: a surface of genus 4. A 3-sheeted covering of the closed surface of genus 2, drawn as 3 copies of its 8-sided face with each side coloured by its generator and numbered with the sheet it glues to. The Euler characteristic −6 is 3 times −2, and the cover has genus 4.

Covering a surface multiplies its count

A covering of a closed surface is a permutation of the sheets for each edge of the surface's one face — with one condition that a covering of a graph never had to meet. When the condition holds, the cells of the cover can be counted directly, and the count is the base's count times the number of sheets. That multiplication decides which surfaces can cover which, before any cover is built.

topology · Covering spaces A + B modulo 13: 4 and 3 residues make 9. A clock face of residues with two sets marked on an inner ring and their sumset marked on an outer ring.

A sum of two sets modulo a prime cannot be small

Add every element of one set of residues to every element of another. Over the whole numbers the sums always number at least |A| + |B| − 1. Modulo a prime the sums can wrap round and collide, and still they never number fewer — the theorem Cauchy proved in 1813 and Davenport again in 1935. Modulo 12 they can. A polynomial of low degree explains the difference in a paragraph.

computation · Finite fields 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 Every pair of rearrangements of four letters, and what the two generate. A 24 by 24 grid of ordered pairs of permutations of four letters: 216 generate S4, 96 generate A4, 264 generate a smaller subgroup.

Two random shuffles reach every shuffle

Pick two ways of rearranging n letters at random and allow them to be repeated in any order. Almost always the two together reach every rearrangement there is, or every even one — and when they fail, the usual reason is a coincidence about a single letter that both happen to leave alone. Eugen Netto guessed it in 1882 and John Dixon proved it in 1969.

algebra · Symmetry groups The 3,600 pairs of icosahedral rotations, and what each pair generates. A 60 by 60 grid of ordered pairs of elements of A5 shaded by the order of the subgroup they generate; 2280 generate A5.

Nineteen copies and no more

Two of the icosahedron's rotations, chosen at random, generate all sixty of them nineteen times in thirty. Philip Hall found the number in 1936 by inclusion and exclusion over every subgroup at once, weighted by the Möbius function of the subgroups' order — and the same count says that nineteen copies of the group, side by side, can still be generated by two elements, while twenty cannot.

algebra · Symmetry groups

Named alongside it

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

Covering spaceFundamental groupIndexFree groupGraphAlternating groupField extensionGalois groupGenerating setLiftingMonodromyProbability

All concepts