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 6 essays across 2 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

Named alongside it

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

Covering spaceIndexFree groupFundamental groupGraphLiftingRankSpanning treeAutomorphismDecision procedureDeck transformationDegree

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