need help...finite direct sums

**deniselim17** · November 1st 2008, 02:49 AM

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 $d(K)$ .

**deniselim17** · November 1st 2008, 05:32 PM

Originally Posted by deniselim17

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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