# Thread: need help...finite direct sums

1. ## 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, 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 $d(K)$.

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