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 2 essays across one field — 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

Named alongside it

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

Alternating group Commutator Conjugacy class Coset Counting argument Cyclic group Derived series Dihedral group Equivalence Group action Lagrange theorem Modular arithmetic

All concepts