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## Finite abelian group into sequence of subgroups

G finite abelian group

WTS: There exist sequence of subgroups {e} = Hr c .... c H1 c G
such that Hi/Hi+1 is cyclic of prime order for all i.

My original thought was to create Hi+1 by reducing the power of one of the generators of Hi by a prime p. Then the order of Hi/Hi+1 would be p, but not necessarily cyclic.

I also know that a simple finite abelian group is cyclic of prime order, but don't know how to construct simple cosets.

(kiHi+1)(hiHi+1)(kiHi+1)-1 = (hiHi+1) where hi c Hi, ki c Ki c Hi

since G abelian implies the inverse coset commutes.

Then, in order to prove Hi/Hi+1 is simple would be equivalent to showing that the trivial subgroup and itself are the only subgroups. If that were true, then there would only be one valid subgroup of G in the sequence. Ie. the sequence would look like {e} c H c G.

What am I missing here?
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