Concept

Abelian group

A group in which the order of combining never matters: a times b always equals b times a. The whole numbers under addition and the symmetries of turning a wheel are examples, and every finite one is a product of cyclic groups.

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

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

Which pairs of the square's symmetries commute: 40 of 64. A 8-by-8 grid over the elements of the 8 symmetries of a 4-gon, rows and columns grouped into 5 conjugacy classes, with a filled square wherever the two elements commute: 40 filled squares, and each row's count of filled squares written at its end.

Five-eighths of the pairs, and no more

Pick two symmetries of a square at random and do them in both orders: forty times in sixty-four the result is the same. No group that fails to commute does better. The reason is a count of pairs that turns into a count of conjugacy classes, and a two-line argument about the centre that caps the answer at five-eighths — reached by the square and the quaternions, approached from above by nothing, and approached from below by groups that commute a little more than half the time.

algebra · Symmetry groups

Named alongside it

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

Commutator Conjugacy class Commutativity Dihedral group Exhaustive search Homomorphism Invariant Normal subgroup Partition Permutation Probability Quaternion

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