# Classifying Finite Abelian Groups

## Homework Statement

Count and describe the different isomorphism classes of abelian groups of order 1800. I don't need to list the group individually, but I need to give some sort of justification.

## The Attempt at a Solution

I'm using the theorem to classify finitely generated abelian groups,
As always we will have Z_1800 to begin with.
Also we know 1800=23(32)(52).

But how do I count all of the possibilities?

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Why don't you start listing them individually? Maybe that will jog your imagination enough to figure out how to count them without listing them.

I've counted all the ones that you get by using the classification theorem for finite abelian groups but a problem I've encountered is using the information given by the Sylow theorems as none of the groups from the first theorem contain the necessary subgroups.
What I've gathered is that there must be at least one subgroup of order 8, 9, and 25 and there a bunch of possibilities for how many for all of them. The numbers just don't really add up right. Any ideas how to approach this?

You don't need the Sylow theorems. You have a complete classification theorem for abelian groups. That's all you need. That certainly gives you subgroups of order 8, 9 and 25. And several other orders besides. I'm not sure what you are fretting about. Which structure from the classification theorem doesn't give you these?