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Is this is true? Let $G=\sum_{i\in I}H_{i}$, such that $H_{i}$ are finite $\bf{cyclic}$ p-groups and $I$ may to be infinite. Now if $T$ to be subgroup of $G$ then do $T=\sum_{i\in I}N_{i}$, such that any $N_{i}$ 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
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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 math.stackexchange.com/questions/13891/… –  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
    
@Ali: Some revisions are needed. –  Babak S. Jan 18 '12 at 8:42

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