Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order

In this paper we formalized some theorems concerning the cyclic groups of prime power order. We formalize that every commutative cyclic group of prime power order is isomorphic to a direct product of family of cyclic groups [1], [18]. Let G be a finite group. The functor Ordset(G) yielding a subset of N is defined by the term (Def. 1) the set of all ord(a) where a is an element of G. One can check that Ordset(G) is finite and non empty. Now we state the propositions: (1) Let us consider a finite group G. Then there exists an element g of G such that ord(g) = sup Ordset(G).