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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 3 essays across 2 fields — 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 A loop that cannot be undone, and the surface it bounds. The loop aba⁻¹b⁻¹ on the figure eight, which cannot be pulled tight, beside a square whose boundary reads the same word, showing that the loop bounds a surface.

What homology forgets about a loop

Let the letters of a loop commute and the loop group of a space becomes its first homology group: a loop now records only how often it went round each hole. What is thrown away is exactly the loops that bound a surface. On the figure eight that is nearly everything — of the loops of sixty letters that homology calls nought, about one in 6,700 is a loop that actually shrinks.

topology · Homotopy

Named alongside it

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

CommutatorConjugacy classCommutativityCovering spaceDihedral groupExhaustive searchFree groupFundamental groupGenusHomologyHomomorphismInvariant

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