00:01
Hello, in this question we have to find all abelian groups of order 1500 up to isomorphism.
00:09
Okay, before we answer that question, answer this question, let us recall the fundamental theorem of finite abelian group on which the solution will depend.
00:18
It is every finite abelian group is isomorphic to a direct product of cyclic groups of prime power order and this direct product is unique up to a rearrangement of the factors.
00:31
That is g is isomorphic to a direct product of cyclic subgroups of prime power order.
00:40
The order of the cyclic subgroup is some prime p1 to the power n1.
00:44
So, g is isomorphic to a direct product like this.
00:47
It is a finite direct product because g is finite and if you multiply the orders it will give a restriction on the number of factors that are possible and this representation is unique up to that is if g is represented like this and it has another representation as a direct product of prime cyclic subgroups cyclic groups of prime power order then the factors then the list of factors will be the same as the list of factors here counting multiplicity.
01:20
The set of factors will be same and their multiplicity in this factorization also will be the same.
01:25
It is just that the arrangement the order in which they are listed in the product they might be different...