Cayley's Theorem

Abstract

The notation and terminology used in this paper have been introduced in the following papers: [3], [6], [4], [5], [10], [11], [7], [2], [1], [9], and [8]. In this paper X, Y denote sets, G denotes a group, and n denotes a natural number. Let us consider X. Note that ∅X,∅ is onto. Let us observe that every set which is permutational is also functional. Let us consider X. The functor permutationsX is defined as follows: (Def. 1) permutationsX = {f : f ranges over permutations of X}. Next we state three propositions: (1) For every set f such that f ∈ permutationsX holds f is a permutation of X. (2) permutationsX ⊆ XX . (3) permutations Seg n = the permutations of n. Let us consider X. One can verify that permutationsX is non empty and functional. Let X be a finite set. One can verify that permutationsX is finite. Next we state the proposition (4) permutations ∅ = 1. Let us consider X. The functor SymGroupX yields a strict constituted functions multiplicative magma and is defined by: