Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $H_i$?

share|cite|improve this question
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

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.