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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.2012-01-18
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    Consider the case when $I=\{1,2\}$ and $H_1=H_2=$some small example.2012-01-18
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    See also http://math.stackexchange.com/questions/13891/finite-groups-h-leq-a-times-b-is-h-cong-c-times-d-for-some-c-leq-a2012-01-18
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    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.2012-01-18

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