Let p be a prime number and consider $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p^2\mathbb{Z}$. How many subgroups of order p does it have? Given any two subgroups $B_1, B_2 $ of $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p^2\mathbb{Z}$ of order p, is there an automorphism $f$ of $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p^2\mathbb{Z}$ such that $f(B_1) =B_2$?
Number of subgroups of order p
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group-theory
abelian-groups
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0@Andrew: I hope it is clear now. – 2012-10-30
1 Answers
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Hint: The subgroups of order $p$ are generated by $(1,pk)$ for $k=0,..,p-1$ and $(0,p)$. That gives $p+1$ distinct subgroups.
Now, find two such subgroups, $B_1$ and $B_2$ such that $G/B_1\not\cong G/B_2$ (where $G$ is your product group.) That would answer your second question in the negative.
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0Great hint! Thanks a lot! – 2012-10-30