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Concept

Lagrange theorem

The rule that the size of any subgroup divides the size of the group containing it. It follows from the cosets all having the same size, and it is what limits which subgroups a group can have at all.

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

Every relabelling of a 4-gon's corners, and the 8 that are motions. All 24 permutations of the corners drawn one by one, with the 8 that preserve every distance marked; the rest deform the polygon and are not symmetries.

Eight ways to leave a square alone

A square can be picked up and put back so that nothing looks different. There are exactly eight ways to do it, and the number is not asserted here — it is what a search through all twenty-four relabellings of the corners comes back with.

algebra · symmetry groups 16 colourings in 6 classes. Every way of colouring the corners, with the ones a motion carries to each other placed on the same row; the number of rows is the number of genuinely different colourings.

Colourings nobody can tell apart

Sixteen ways to colour four corners in two colours, and only six of them are genuinely different. The count can be got by pooling the sixteen — or by never forming a single class and instead averaging how many colourings each motion leaves untouched.

algebra · symmetry groups 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

Named alongside it

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

Counting argumentCyclic groupDihedral groupGroup actionEquivalencePermutationSymmetryClosureCosetCounting two waysInvariantModular arithmetic

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