Concept

Group — where it appears

A set with a way of combining any two of its elements that is associative, has an identity, and gives every element an inverse. It is the standard object for describing symmetry, because the motions that leave something unchanged always form one.

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

The permutation (1 3 4 2) drawn as 4 strings, crossing 3 times. A permutation drawn as strings running from a row of numbered pegs to another, with every place two strings cross marked, and the crossing count checked against the number of pairs that are out of order.

The crossings that will not come out even

Draw a rearrangement as strings from one row of pegs to another and count where they cross. The count depends on how the strings are drawn; whether it is odd or even does not, and that single bit is what makes determinants exist and a sliding puzzle unsolvable.

algebra · permutation parity

The quaternion multiplication table, from i² = j² = k² = ijk = −1. A four-by-four multiplication table of the quaternion units, with the row giving the left factor, every entry computed from Hamilton's rule, and the pair that differs between the two orders marked.

A multiplication that remembers the order

Four characters — i² = j² = k² = ijk = −1 — define a multiplication in which ab and ba are different numbers. Everything follows from them, including the fact that a rotation of a solid body has two names, and that two full turns are needed to get one of them home.

algebra · quaternions

Named alongside it

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

Symmetry Associativity Bijection Complex numbers Conjugate Counting argument Determinant Dimension Division algebra Invariant Multiplication Parity

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