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Let $G = \sum_{i\in I}H_i$ where $H_i$ are finite cyclic $p$-groups and $I$ may be infinite. Let $T$ be a subgroup of $G$. Is it true that $T = \sum_{i\in I}N_{i}$ where $N_i$ is normal subgroup of $H_i$?

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All subgroups of an abelian group are normal, so there is no need to include that adjective. –  Jonas Meyer Jan 18 '12 at 7:16
Consider the case when $I=\{1,2\}$ and $H_1=H_2=$some small example. –  Jonas Meyer Jan 18 '12 at 7:19
See also… –  Jonas Meyer Jan 18 '12 at 7:21
I noticed you just added the adjective "cyclic", but this doesn't change the problem, because every finite abelian group is already a direct sum of cyclic groups. In any case I recommend that for your counterexample you consider $2$ cyclic groups. –  Jonas Meyer Jan 18 '12 at 7:39

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This is not true. Consider $G = \mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$ and the subgroup $T = \langle (1, 1)\rangle \cong \mathbb{Z}/2\mathbb{Z}$.

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