23cyclic

23cyclic is a term used in mathematics and theoretical computer science to describe structures that exhibit a cyclic symmetry of order 23. The name combines the number 23 with the concept of cyclicity, indicating that the relevant symmetry repeats after 23 steps or positions.

Definition: A set with a 23cyclic structure supports an action of the cyclic group of order 23, denoted C23, generated by a permutation that cycles elements in a fixed sequence. Many 23cyclic objects have a transitive C23-action, meaning that a single application of the generator moves any element to any other within the same orbit.

Examples: A permutation on n elements that is a 23-cycle is a simple 23cyclic object. A time-series with period 23 is 23cyclic in the sense that values repeat every 23 steps. In graph theory, a 23cyclic graph is one that is invariant under a 23-step rotation of its vertices in a symmetric embedding, and whose automorphism group contains a cyclic subgroup of order 23.

Properties and theory: Because 23 is prime, C23 is cyclic and has no nontrivial proper subgroups; this constrains the structure of 23cyclic objects. Counting such structures often uses Burnside's lemma or orbit-stabilizer arguments. In coding and cryptography, cyclic groups of prime order underpin certain schemes and are sometimes described as 23cyclic in informal literature.

See also: cyclic group, permutation, orbit, Burnside's lemma, discrete logarithm.