Proving converse of fundamental theorem of cyclic groups

by curiousmuch
Tags: converse, cyclic, fundamental, groups, proving, theorem
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 P: 9 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
 Emeritus Sci Advisor PF Gold P: 5,532 Post what you've done on this problem please.
 Sci Advisor HW Helper P: 9,396 Can you check the question. C_2 x C_2 has subgroups of orders 1,2 and 4, but is not cyclic.
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Proving converse of fundamental theorem of cyclic groups

 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.
 Sci Advisor HW Helper P: 9,396 And that's why we have the word 'exactly'.

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