Proving converse of fundamental theorem of cyclic groups

[curiousmuch]

Homework Statement

If G is a finite abelian group that has one subgroup of order d for every divisor d of the order of G. Prove that G is cyclic.

Homework Equations

The Attempt at a Solution

 

[quantumdude]

Post what you've done on this problem please.

 

[matt grime]

Can you check the question. C_2 x C_2 has subgroups of orders 1,2 and 4, but is not cyclic.

 

[Dick]

Can you check the question. C_2 x C_2 has subgroups of orders 1,2 and 4, but is not cyclic.

I think the point is that it is supposed to have ONE subgroup of each order. Your example has several subgroups of order 2.

 

[matt grime]

And that's why we have the word 'exactly'.