Canonical Description of Group Theory: A Linear Order on All Finite Groups

We provide an axiomatic base for the set of natural numbers, that has been proposed as a canonical construction, and use this definition of \(\mathbb N\) to find several results on finite group theory. Every finite group \(G\), is well represented with a natural number \(N_G\); if \(N_G=N_H\) then $H,G$ are in the same isomorphism class. We have a linear order, on the quotient space of isomorphism classes of finite groups, that is well behaved with respect to cardinality. If $H,G$ are two finite groups such that $|H|=m<n=|G|$, then \(H<\mathbb Z_n\leq G\leq\mathbb Z_{p_1}^{n_1}\oplus\mathbb Z_{p_2}^{n_2}\oplus\cdots\oplus\mathbb Z_{p_k}^{n_k}\) where \(n=p_1^{n_1}p_2^{n_2}\cdots p_{k}^{n_k}\) is the prime factorization of \(n\). We find a canonical order for the objects of \(G\) and define equivalent objects of \(G\), thus finding the automorphisms of \(G\). The Cayley table of \(G\) takes canonical block form, and we are provided with a minimal set of independent equations that define the group. We show how to find all groups of order \(n\), and order them. We give examples using all groups with order smaller than $10$, and we find the canonical block form of the symmetry group \(\Delta_4\). In the next section, we extend our results to the infinite case, which defines a real number as an infinite set of natural numbers. A real function is a set of real numbers, and a sequence of real functions \(f_1,f_2,\ldots\) is well represented by a set of real numbers, as well. In general, we represent mathematical objects using the smallest possible data-type. In the last section, mathematical objects are well assigned to tree structures. We conclude with brief comments on type theory and future work on computational aspects of these representations.