How many mutually nonisomorphic Abelian groups of order 50

[Whenry]

Homework Statement

My question is about the rules behind the method of finding the solution not necessarily the method itself.

(I am prepping for the GRE subject)

How many mutually nonisomorphic Abelian groups of order 50?

The Attempt at a Solution

so, if understand this correctly, we have 2 of these groups:

1) Z2 + Z5 + Z5

2) Z2 + Z25

However, what I don't understand is why these are mutually nonisomorphic. The theorems presented before this problem state:

The direct sum Zm + Zn is cyclic iff gcd(m,n) = 1. If this is the case, then, since Zm + Zn has order mn, Zm +Zn is isomorphic to Zmn,

So are groups 1 and 2 both isomorphic to Z50 but not isomorphic to each other? which doesn't make sense to me.

 

[micromass]

Homework Statement

My question is about the rules behind the method of finding the solution not necessarily the method itself.

(I am prepping for the GRE subject)

How many mutually nonisomorphic Abelian groups of order 50?

The Attempt at a Solution

so, if understand this correctly, we have 2 of these groups:

1) Z2 + Z5 + Z5

2) Z2 + Z25

However, what I don't understand is why these are mutually nonisomorphic. The theorems presented before this problem state:

The direct sum Zm + Zn is cyclic iff gcd(m,n) = 1. If this is the case, then, since Zm + Zn has order mn, Zm +Zn is isomorphic to Zmn,

So are groups 1 and 2 both isomorphic to Z50 but not isomorphic to each other? which doesn't make sense to me.

No, not at all. Only group (2) is isomorphic to Z50. Group (1) is not cyclic.

How do we prove that (1) and (2) are not isomorphic?? Well, the easiest way is to check the orders of the elements. Prove that (2) has an element of order 25, but (1) does not have such an elements.

 

[Whenry]

No, not at all. Only group (2) is isomorphic to Z50. Group (1) is not cyclic.

How do we prove that (1) and (2) are not isomorphic?? Well, the easiest way is to check the orders of the elements. Prove that (2) has an element of order 25, but (1) does not have such an elements.

Ok, yes, it makes sense now. For some reason I ignored the gcd(m,n). Thank you.