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Every abelian $p$-group is isomorphic to a direct sum of cyclic $p$-groups.

We have that every abelian $p$-group is an image of some direct sum of cyclic $p$- groups. Therefore, every abelian $p$-group is a quotient of the direct sum of the family of cyclic $p$-groups. Now, the quotient of the direct sum of the family of cyclic $p$-groups is direct sum of the family of cyclic p-groups (I am not sure this is correct). Hence every abelian $p$-group is isomorphic to some direct sum of cyclic $p$-groups

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    I tried to fix the spelling, but I don't follow the penultimate sentence. Maybe it would help to give names to things? Are you saying that the quotient is a direct _summand_ of the direct sum?2012-01-18
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    Your use of "the" makes this problematic. While it is true that a quotient of a direct sum of a given family of cyclic $p$-group is a direct sum of *some* family of cyclic $p$-groups, this family need not be the original one. For instance, $C_2\oplus C_2$ is a quotient of $C_4\oplus C_4$, but the family involved in the latter consists of two copies of $C_4$ and the family involved in the former of two copies of $C_2$. It is also unclear on what grounds you make the jump from "quotient of a direct sum of cyclic $p$-groups" to "direct sum of cyclic $p$-groups". That's the meat of the problem!2012-01-18
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    The structure theorem for finite abelian groups says they're all direct sums of cyclic groups. If you're allowed to use that, you should have no difficulty with the question in the title.2012-01-18
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    @Gerry: That's only applicable to finitely generated abelian $p$-groups. Of course, it all depends on whether "$p$-groups" means "group whose order is a power of $p$" or "group all of whose elements have order a power of $p$".2012-01-18
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    @Ali Gholamian: By "$p$-group" do you mean "finite group whose order is a power of $p$", or do you mean "group all of whose elements have order a power of $p$"? The result may not be true in the latter case if your group is not finitely generated.2012-01-18
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    p- groups is "group all of whose elements have order a power of p'2012-01-18
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    @Arturo, I always take as my default assumption that all groups are finite. OP can always disabuse me of my misperception.2012-01-18

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