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How can we decide how a group is decomposed as a direct sum of cyclic $p$-groups from the character table? Assume the group is finite abelian and that we know the complex character table.

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You should digest the answer to your previous question first. Then you wouldn't need to ask this one. –  Alex B. Jun 19 '12 at 8:03
    
The rows of the character table of a finite abelian group $G$ are elements of the group $(\mathbb{C}^\times)^{|G|}$. They form a group under multiplication. That group is isomorphic to $G$. –  Jack Schmidt Jun 19 '12 at 13:34
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The number of irreducible characters will tell you the order of the group. Then, for each prime $p$ dividing the order of the group, and each $r$, the number of characters of order $p^r$ gives you the number of group elements of order $p^r$. And once you know the number of elements of each order, you can get the decomposition into a direct sum of cyclic $p$-groups.

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