Concept

Normal subgroup

A subgroup carried into itself by conjugation with every element of the group. It is exactly the condition under which the cosets can be multiplied, so it is what makes a quotient group exist, and a chain of them with abelian steps is what solvability by radicals requires.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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 derived series of S3, S4, S5. A table with one row per group giving the sizes along its derived series, each step the subgroup generated by all commutators of the last, and whether the series reaches the identity.

The group that will not come apart

Solving an equation by radicals means building a tower of roots, and a tower of roots corresponds to a chain of subgroups with abelian steps. For the general equation of degree five that chain would have to descend through a group of sixty elements with no normal subgroup in it — so there is no formula, and the obstruction is a finite object that can be written out.

algebra · Galois correspondence

The half of 24 permutations that commutators reach. A block of 24 squares, one per permutation, with the 12 generated by commutators shaded, beside bars counting the homomorphisms to each cyclic group.

The only bit that survives

A shuffle can be called even or odd, and the label behaves under composition. Ask whether some cleverer label — a number out of three, or out of four — could behave the same way, and the answer is that nothing else can — one bit is exactly what a permutation gives up.

algebra · Permutation parity

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

The three pairings of the roots of x⁴ + x + 1. Three panels each showing the same four roots of a quartic in the complex plane, joined in a different way into two pairs, with the value of the sum of the pair products beneath.

Three ways to pair four roots

Four roots can be split into two pairs in exactly three ways, and the three numbers r·r′ + r″·r‴ those pairings give are the roots of a cubic whose coefficients can be read straight off the quartic. That cubic is where Ferrari's formula gets its cube roots, and it is also a verdict: whether its roots are rational decides which of the twenty-four symmetries the quartic's roots actually have.

algebra · Galois correspondence

Named alongside it

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

Commutator Conjugacy class Permutation Abelian group Alternating group Coset Counting argument Covering space Cyclic group Deck transformation Derived series Dihedral group

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