Proving converse of fundamental theorem of cyclic groups

curiousmuch

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. The attempt at a solution

Tom Mattson

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

Quote by matt grime

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'.

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curiousmuch	Proving converse of fundamental theorem of cyclic groups Tweet 1. The problem statement, all variables and given/known data 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. 2. Relevant equations 3. The attempt at a solution

Tom Mattson Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus	Post what you've done on this problem please.

matt grime Recognitions: Homework Help Science Advisor	Can you check the question. C_2 x C_2 has subgroups of orders 1,2 and 4, but is not cyclic.