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Math Help - need help...finite direct sums

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

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

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

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

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

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

    Now, my question in no. 2 is "If n<k, then what is p^{n}G?"
    Is it 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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