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Thread: need help...finite direct sums

  1. #1
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    need help...finite direct sums

    1. Let $\displaystyle G=\sigma(m_{1}) \oplus ... \oplus \sigma(m_{s})$ be a canonical decomposition. Show that $\displaystyle \mid G \mid = \coprod m_{i}$ and that $\displaystyle m_{s}$ is the least positive integer $\displaystyle n$ for which $\displaystyle nG=\{{0}\}$.

    2. Let $\displaystyle G$ be a direct sum of $\displaystyle b$ copies of cyclic groups of order $\displaystyle p^{k}$. If $\displaystyle n<k$, then $\displaystyle d(p^{n}G/p^{n+1}G)=b$.

    3. If $\displaystyle H$ and $\displaystyle K$ are elementary $\displaystyle p$-primary abelian groups, then $\displaystyle d(H \oplus K)=d(H)+d(K)$, where $\displaystyle d(H)$ is the number of cyclic summands occuring in a decomposition of an elementary abelian group depends only on $\displaystyle H$ and similar for $\displaystyle d(K)$.
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  2. #2
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    Quote Originally Posted by deniselim17 View Post
    1. Let $\displaystyle G=\sigma(m_{1}) \oplus ... \oplus \sigma(m_{s})$ be a canonical decomposition. Show that $\displaystyle \mid G \mid = \coprod m_{i}$ and that $\displaystyle m_{s}$ is the least positive integer $\displaystyle n$ for which $\displaystyle nG=\{{0}\}$.

    2. Let $\displaystyle G$ be a direct sum of $\displaystyle b$ copies of cyclic groups of order $\displaystyle p^{k}$. If $\displaystyle n<k$, then $\displaystyle d(p^{n}G/p^{n+1}G)=b$.

    3. If $\displaystyle H$ and $\displaystyle K$ are elementary $\displaystyle p$-primary abelian groups, then $\displaystyle d(H \oplus K)=d(H)+d(K)$, where $\displaystyle d(H)$ is the number of cyclic summands occuring in a decomposition of an elementary abelian group depends only on $\displaystyle H$ and similar for $\displaystyle d(K)$.
    I had shown no. 1.

    Now, my question in no. 2 is "If $\displaystyle n<k$, then what is $\displaystyle p^{n}G$?"
    Is it $\displaystyle p^{n}G=p^{n}B_{n+1} \oplus p^{n}B_{n+2} \oplus ... \oplus p^{n}B_{n+k}$?

    And I think no. 3 I can show it.
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