Isomorphisms of Direct Products of Finite Commutative Groups

Summary We have been working on the formalization of groups. In [1], we encoded some theorems concerning the product of cyclic groups. In this article, we present the generalized formalization of [1]. First, we show that every finite commutative group which order is composite number is isomorphic to a direct product of finite commutative groups which orders are relatively prime. Next, we describe finite direct products of finite commutative groups


Preliminaries
Now we state the propositions:   (3) If support p 2 misses support p 1 and f = p 2 + p 1 and q ∈ support p 2 , then p 2 (q) = f (q). (4) If support p 2 misses support p 1 and f = p 2 + p 1 and q ∈ support p 1 , then p 1 (q) = f (q). Now we state the propositions: (5) Let us consider a non zero natural number h and a prime number q. If q and h are not relatively prime, then q | h. (6) Let us consider non zero natural numbers h, s. Suppose a prime number q. Suppose q ∈ support PrimeFactorization(s). Then q and h are not relatively prime. Then support PrimeFactorization(s) ⊆ support PrimeFactorization(h). The theorem is a consequence of (5). (7) Let us consider non zero natural numbers h, k, s, t. Suppose (i) h and k are relatively prime, and (ii) s · t = h · k, and (iii) for every prime number q such that q ∈ support PrimeFactorization(s) holds q and h are not relatively prime, and (iv) for every prime number q such that q ∈ support PrimeFactorization(t) holds q and k are not relatively prime.

Then
(v) s = h, and The theorem is a consequence of (6), (1), (3), and (4). Proof: Set p 2 = PrimeFactorization(s). Set p 1 = PrimeFactorization(t). For every natural number p such that p ∈ support PFExp(h) holds p 2 (p) = p p -count(h) . For every natural number p such that p ∈ support PFExp(k) holds p 1 (p) = p p -count(k) . Let G be a non empty multiplicative magma, I be a finite set, and b be a (the carrier of G)-valued total I-defined function. The functor b yielding an element of G is defined by (Def. 1) There exists a finite sequence f of elements of G such that Then F 1 = F 3 · F 2 . (9) Let us consider a non empty multiplicative magma G, a set q, an element z of G, and a (the carrier of G)-valued total {q}-defined function f . If f = q −→ . z, then f = z.

Direct Product of Finite Commutative Groups
Now we state the propositions: (10) Let us consider non empty multiplicative magmas X, Y . Then the carrier of X, Y = the carrier of X, the carrier of Y . Proof: Set CarrX = the carrier of X. Set CarrY = the carrier of Y . For every element a such that a ∈ dom the support of X, Y holds (the support of X, Y )(a) = the carrier of X, the carrier of Y (a). Then there exists a homomorphism h from A, B to G such that (iii) h is bijective, and The theorem is a consequence of (11 Let us consider a finite commutative group G, a natural number m, and a subset A of G. Now we state the propositions: The theorem is a consequence of (14).  every elements a, b of G such that a ∈ H and The theorem is a consequence of (16), (12), (17), (15), and (7).

Finite Direct Products of Finite Commutative Groups
Let us consider a group G, a set q, an associative group-like multiplicative magma family F of {q}, and a function f from G into F . Now we state the propositions: (19) If F = q −→ . G and for every element x of G, f (x) = q −→ .
x, then f is a homomorphism from G to F . The theorem is a consequence of (19) and (20).   The theorem is a consequence of (21), (25), and (9). Proof: Consider I being a homomorphism from G to F such that I is bijective and for every element The theorem is a consequence of (27). Proof: Set L0 = G 0 −1 . Consider L being a homomorphism from H, K to F such that L is bijective and for every element h of H and for every element k of K, there exists a function g such that g = L0(h) and L( h, k ) = g+·(q −→ . k). Set G = L −1 . For every function x 0 and for every element k of K and for every element h of H such that h = G 0 (x 0 ) and x 0 ∈ F 0 holds G(x 0 +·(q −→ . k)) = h, k .
Then there exists a total I 0 -defined function x 0 and there exists an element k of K such that x 0 ∈ F 0 and x = x 0 +·(q −→ . k) and for every element p of I 0 , x 0 (p) ∈ F 0 (p). Proof: Reconsider y = x as a total I-defined function. Reconsider k = y(q) as an element of K. Reconsider y0 = y I 0 as an I 0 -defined function. For every element i of I 0 , y0(i) ∈ (the support of F 0 )(i) and y0(i) ∈ F 0 (i).
Then x = x 0 · k. The theorem is a consequence of (8) and (9). Proof: Reconsider y = q −→ . k as a (the carrier of G)-valued total {q}-defined function. I 0 misses {q}. Let us consider a finite commutative group G. Now we state the propositions: (34) Suppose G > 1. Then there exists a non empty finite set I and there exists an associative group-like commutative multiplicative magma family F of I and there exists a homomorphism H 0 from F to G such that I = support PrimeFactorization(G) and for every element p of I, F (p) is a subgroup of G and F (p) = (PrimeFactorization(G))(p) and for every elements p, q of I such that p = q holds (the carrier of F (p)) ∩ (the carrier of F (q)) = {1 G } and H 0 is bijective and for every (the carrier of G)-valued total I-defined function x such that for every element p of I, x(p) ∈ F (p) holds x ∈ F and H 0 (x) = x.
(35) Suppose G > 1. Then there exists a non empty finite set I and there exists an associative group-like commutative multiplicative magma family F of I such that I = support PrimeFactorization(G) and for every element p of I, F (p) is a subgroup of G and F (p) = (PrimeFactorization(G))(p) and for every elements p, q of I such that p = q holds (the carrier of F (p)) ∩ (the carrier of F (q)) = {1 G } and for every element y of G, there exists a (the carrier of G)-valued total I-defined function x such that for every element p of I, x(p) ∈ F (p) and y = x and for every (the carrier of G)-valued total I-defined functions x 1 , x 2 such that for every element p of I, x 1 (p) ∈ F (p) and for every element p of I, x 2 (p) ∈ F (p) and x 1 = x 2 holds x 1 = x 2 .