Cohomology and Extensions
A permutation is an automorphism in Set. More explicitly, a permutation of a set is an invertible function from to itself.
The group of permutations of (that is the automorphism group of in ) is the symmetric group (or permutation group) on . This group may be denoted , , or . When is the finite set with elements, then its symmetric group is a finite group, and one typically writes or ; note that this group has elements.
In combinatorics, one often wants a slight generalisation. Given a natural number , an -permutation of is an injective function from to , that is a list of distinct elements of . Note that the number of -permutations of is counted by the falling factorial . Then an -permutation of is the same as a permutation of , and the total number of such permutations is of course counted by the ordinary factorial . (That an injective function from to itself must be invertible characterises as a Dedekind-finite set.)
Via string diagrams
Via cycle decompositions
Every permutation generates a cyclic subgroup of the symmetric group , and hence inherits a group action on . The orbits of this action partition the set into a disjoint union of cycles, called the cyclic decomposition of the permutation .
For example, let be the permutation on defined by
The domain of the permutation is partitioned into three -orbits
corresponding to the three cycles
Finally, we can express this more compactly by writing in cycle form, as the product .
Relation to the field with one element
One may regard the symmetric group as the general linear group in dimension on the field with one element .
Whitehead tower and relation to supersymmetry
The symmetric groups and alternating groups are the first stages in a restriction of the Whitehead tower of the orthogonal group to “finite discrete ∞-groups” in the sense of homotopy type with finite homotopy groups. The homotopy fibers of the stages of the “finite Whitehead tower” are the stable homotopy groups of spheres (Epa-Ganter 16). (See also at super algebra – Abstract idea and at Platonic 2-group.)
Notice that the squares on the right are not homotopy pullback squares. (The homotopy pullback of the string 2-group along is a -extension of , but here we get the universal finite 2-group extension, by instead.