Fixedpointfree

Fixed-point-free describes a property of a map or action on a set or space, where no element is left fixed. Formally, a function f: X → X is fixed-point-free if f(x) ≠ x for all x ∈ X. The term is used across several branches of mathematics, with context determining its exact meaning.

In combinatorics, a fixed-point-free permutation of a finite set is called a derangement. Such a permutation has no 1-cycles in its cycle decomposition. The number of derangements of n elements is !n = n! ∑_{k=0}^n (-1)^k/k!. For example, !4 = 9.

In group theory, a group action of G on a set X is fixed-point-free (or free) if only the identity element fixes any point: for all g ≠ e and x ∈ X, g·x ≠ x. This occurs in regular actions, where each orbit has size |G| and stabilizers are trivial. Derangements arise as fixed-point-free elements in permutation representations.

In topology and dynamics, fixed-point-free maps interact with fixed-point theorems. Brouwer's fixed-point theorem implies that every continuous map from a closed disk to itself has a fixed point, so the disk admits no fixed-point-free continuous self-map. On other spaces, fixed-point-free maps can exist; for example, a rotation of the circle S^1 by a nonzero angle has no fixed points on S^1.

Applications and theory: In finite group theory, fixed-point-free automorphisms (those with no nontrivial fixed points) are studied for structural consequences of the group; classic results relate such automorphisms to nilpotence or solvability in the group.