Generating set
Named by 4 essays across one field — each of them below, with the objects they name alongside it.
The group drawn as a map
A multiplication table says everything about a group and shows nothing; lay the same information out as one dot per element and one arrow per generator, and multiplying becomes walking, distance becomes a word length, and the group acquires a shape.
How fast the ball fills
Count the elements within r steps of doing nothing. The count grows like a polynomial in some groups and like a power of three in others, the distinction survives every change of generating set, and which polynomial degrees are possible is a theorem nobody expected.
Two random shuffles reach every shuffle
Pick two ways of rearranging n letters at random and allow them to be repeated in any order. Almost always the two together reach every rearrangement there is, or every even one — and when they fail, the usual reason is a coincidence about a single letter that both happen to leave alone. Eugen Netto guessed it in 1882 and John Dixon proved it in 1969.
Nineteen copies and no more
Two of the icosahedron's rotations, chosen at random, generate all sixty of them nineteen times in thirty. Philip Hall found the number in 1936 by inclusion and exclusion over every subgroup at once, weighted by the Möbius function of the subgroups' order — and the same count says that nineteen copies of the group, side by side, can still be generated by two elements, while twenty cannot.
Named alongside it
The objects these essays reach for when they reach for this one.
Alternating groupCayley graphCyclic groupGroup actionProbabilitySubgroupWord metricCosetCycleDihedral groupFree groupGrowth rate