Transitivity

Stick-stuck approach

 * Sets : Stick one stuck none {X,X,.....,X}
 * 15-puzzle : Stick one stuck none
 * Rotation polyhedra : Stick-1-stuck-1
 * Rukik's edges: Stick-1-stuck-2
 * Rukik's corners : Stick-1-stuck-3
 * Groups, Cycles, Cylinders : Stick-1-stuck-ALL (X,X,X,X,...,X)
 * Fields and geometrical lines : Stick-2-stuck-ALL
 * Projective lines : Stick-3-stuck-ALL

The spectrum of species

 * Applies to almost any power series of enumeration, which is a more basic concept than this one. Even so, I don't think the table adds anything.  There are three examples above (sets, permutations, pairs), which seem adequate. AR

Carmichael 1937
Every possible algebra A[s] is an algebra A[pn] defined as in $102, by means of a doubly transitive group of prime-power degree pn and order pn(pn-1), while, conversely, such a doubly transitive group is induced by the totality of transformations of the form (2) on the marks of such an algebra.

Carmichael, Groups of finite order, 1937, page 401

There is a deep identity between


 * 1) doubly transitive groups, in which two points determine the entire permutation
 * 2) algebras (which are here not necessarily commutative fields), that have O and 1
 * 3) geometrical lines, determined by two points and getting coordinates by field numbers.

The Carmichael connection is absolutely natural, given two subgroups - the translations and the rotations in a doubly transitive group.

Using the species notations, we have F" = X.X....X. Stick two, stuck all.