Isomorphisms of Direct Products of Finite Cyclic Groups

Summary In this article, we formalize that every finite cyclic group is isomorphic to a direct product of finite cyclic groups which orders are relative prime. This theorem is closely related to the Chinese Remainder theorem ([18]) and is a useful lemma to prove the basis theorem for finite abelian groups and the fundamental theorem of finite abelian groups. Moreover, we formalize some facts about the product of a finite sequence of abelian groups.

Let G be an Abelian add-associative right zeroed right complementable non empty additive loop structure. Note that G is non empty and Abelian group yielding as a finite sequence.
Let G, F be Abelian add-associative right zeroed right complementable non empty additive loop structures. Note that G, F is non empty and Abelian group yielding as a finite sequence.
We now state the proposition (1) Let X be an Abelian group. Then there exists a homomorphism I from X to X such that I is bijective and for every element x of X holds Let G, F be non empty Abelian group yielding finite sequences. Note that G F is Abelian group yielding.
One can prove the following propositions: (2) Let X, Y be Abelian groups. Then there exists a homomorphism I from X × Y to X, Y such that I is bijective and for every element x of X and for every element y of Y holds I(x, y) = x, y .
(3) Let X, Y be sequences of groups. Then there exists a homomorphism I from X × Y to (X Y ) such that (i) I is bijective, and (ii) for every element x of X and for every element y of Y there exist finite sequences x 1 , y 1 such that x = x 1 and y = y 1 and I(x, y) = x 1 y 1 . (4) Let G, F be Abelian groups. Then (i) for every set x holds x is an element of G, F iff there exists an element x 1 of G and there exists an element x 2 of F such that x = x 1 , x 2 , (ii) for all elements x, y of G, F and for all elements x 1 , y 1 of G and for all elements G, F and for every element x 1 of G and for every element (5) Let G, F be Abelian groups. Then (i) for every set x holds x is an element of G × F iff there exists an element x 1 of G and there exists an element x 2 of F such that x = x 1 , x 2 , (ii) for all elements x, y of G × F and for all elements x 1 , y 1 of G and for all elements x 2 , y 2 of F such that x = x 1 , x 2 and y = y 1 , y 2 holds x + y = x 1 + y 1 , x 2 + y 2 , (iii) 0 G×F = 0 G , 0 F , and (iv) for every element x of G × F and for every element x 1 of G and for every element Let G, H, I be groups, h be a homomorphism from G to H, and h 1 be a homomorphism from H to I. Then h 1 · h is a homomorphism from G to I. Let G, H, I be groups, let h be a homomorphism from G to H, and let h 1 be a homomorphism from H to I. Then h 1 · h is a homomorphism from G to I.
One can prove the following propositions: (7) Let G, H be groups and h be a homomorphism from G to H. If h is bijective, then h −1 is a homomorphism from H to G. (8) Let X, Y be sequences of groups. Then there exists a homomorphism I from X, Y to (X Y ) such that (i) I is bijective, and (ii) for every element x of X and for every element y of Y there exist finite sequences x 1 , y 1 such that x = x 1 and y = y 1 and I( x, y ) = x 1 y 1 . (9) Let X, Y be Abelian groups. Then there exists a homomorphism I from X × Y to X × Y such that I is bijective and for every element x of X and for every element y of Y holds I(x, y) = x, y . (10) Let X be a sequence of groups and Y be an Abelian group. Then there exists a homomorphism I from X × Y to (X Y ) such that (i) I is bijective, and (ii) for every element x of X and for every element y of Y there exist finite sequences x 1 , y 1 such that x = x 1 and y = y 1 and I(x, y) = x 1 y 1 . (11) Let n be a non zero natural number. Then the additive loop structure of (Z R n ) is non empty, Abelian, right complementable, add-associative, and right zeroed.
Let n be a natural number. The functor Z/nZ yields an additive loop structure and is defined by: (Def. 1) Z/nZ = the additive loop structure of (Z R n ). Let n be a non zero natural number. Observe that Z/nZ is non empty and strict.
Let n be a non zero natural number. Note that Z/nZ is Abelian, right complementable, add-associative, and right zeroed.
Next we state a number of propositions: (12) Let X be a sequence of groups, x, y, z be elements of X, and x 1 , y 1 , z 1 be finite sequences. Suppose x = x 1 and y = y 1 and z = z 1 . Then z = x + y if and only if for every element j of dom X holds z 1 (j) = (the addition of X(j))(x 1 (j), y 1 (j)). (13) For every CR-sequence m and for every natural number j and for every integer x such that j ∈ dom m holds x mod m mod m(j) = x mod m(j). (14) Let m be a CR-sequence and X be a sequence of groups. Suppose len m = len X and for every element i of N such that i ∈ dom X there exists a non zero natural number m 1 such that m 1 = m(i) and X(i) = Z/m 1 Z . Then there exists a homomorphism I from Z/( m)Z to X such that for every integer x if x ∈ the carrier of Z/( m)Z, then I(x) = mod(x, m). (15) Let X, Y be non empty sets. Then there exists a function I from X × Y into X × Y such that I is one-to-one and onto and for all sets x, y such that x ∈ X and y ∈ Y holds I(x, y) = x, y .
(16) For every non empty set X holds X = X . (17) Let X be a non-empty non empty finite sequence and Y be a non empty set. Then there exists a function I from X × Y into (X Y ) such that (i) I is one-to-one and onto, and (ii) for all sets x, y such that x ∈ X and y ∈ Y there exist finite sequences x 1 , y 1 such that x = x 1 and y = y 1 and I(x, y) = x 1 y 1 . (18) Let m be a finite sequence of elements of N and X be a non-empty non empty finite sequence. Suppose len m = len X and for every element i of N such that i ∈ dom X holds X(i) = m(i). Then X = m. (19) Let m be a CR-sequence and X be a sequence of groups. Suppose len m = len X and for every element i of N such that i ∈ dom X there exists a non zero natural number m 1 such that m 1 = m(i) and X(i) = Z/m 1 Z . Then the carrier of X = m. (20) Let m be a CR-sequence, X be a sequence of groups, and I be a function from Z/( m)Z into X. Suppose that (i) len m = len X, (ii) for every element i of N such that i ∈ dom X there exists a non zero natural number m 1 such that m 1 = m(i) and X(i) = Z/m 1 Z, and (iii) for every integer x such that x ∈ the carrier of Z/( m)Z holds I(x) = mod(x, m).
Then I is one-to-one. (21) Let m be a CR-sequence and X be a sequence of groups. Suppose len m = len X and for every element i of N such that i ∈ dom X there exists a non zero natural number m 1 such that m 1 = m(i) and X(i) = Z/m 1 Z . Then there exists a homomorphism I from Z/( m)Z to X such that I is bijective and for every integer x such that x ∈ the carrier of Z/( m)Z holds I(x) = mod(x, m).