Can I determine a $p$-group by the number of subgroups
$begingroup$
Let $G$ and $H$ be finite $p$-groups of order $p^n$ where $p$ is a prime. Assume that for every $1leq ileq n$, the number of subgroups of $G$ and $H$ of order $p^i$ are the same. Can I deduce that $G$ and $H$ are isomorphic?
It seems like it works for $n=1,2,3$ but these cases are more or less trivial. For $n>3$, examples become too complicated for me to investigate.
abstract-algebra group-theory finite-groups
asked Jan 20 at 1:40
LeventLevent
2,729925
$endgroup$
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$begingroup$
Let $G$ and $H$ be finite $p$-groups of order $p^n$ where $p$ is a prime. Assume that for every $1leq ileq n$, the number of subgroups of $G$ and $H$ of order $p^i$ are the same. Can I deduce that $G$ and $H$ are isomorphic?
It seems like it works for $n=1,2,3$ but these cases are more or less trivial. For $n>3$, examples become too complicated for me to investigate.
abstract-algebra group-theory finite-groups
asked Jan 20 at 1:40
LeventLevent
2,729925
$endgroup$
add a comment |
$begingroup$
Let $G$ and $H$ be finite $p$-groups of order $p^n$ where $p$ is a prime. Assume that for every $1leq ileq n$, the number of subgroups of $G$ and $H$ of order $p^i$ are the same. Can I deduce that $G$ and $H$ are isomorphic?
It seems like it works for $n=1,2,3$ but these cases are more or less trivial. For $n>3$, examples become too complicated for me to investigate.
abstract-algebra group-theory finite-groups
asked Jan 20 at 1:40
LeventLevent
2,729925
$endgroup$
Let $G$ and $H$ be finite $p$-groups of order $p^n$ where $p$ is a prime. Assume that for every $1leq ileq n$, the number of subgroups of $G$ and $H$ of order $p^i$ are the same. Can I deduce that $G$ and $H$ are isomorphic?
It seems like it works for $n=1,2,3$ but these cases are more or less trivial. For $n>3$, examples become too complicated for me to investigate.
abstract-algebra group-theory finite-groups
abstract-algebra group-theory finite-groups
asked Jan 20 at 1:40
LeventLevent
2,729925
asked Jan 20 at 1:40
LeventLevent
2,729925
asked Jan 20 at 1:40
LeventLevent
2,729925
asked Jan 20 at 1:40
LeventLevent
2,729925
asked Jan 20 at 1:40
LeventLevent
2,729925
2,729925
add a comment |
add a comment |
1 Answer
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$begingroup$
No of course you cannot deduce that? Why would expect to? (Counterexample: $mathtt{SmallGroup}(16,i)$ for $i=2$ and $4$; or for $i=5$ and $6$.)
answered Jan 20 at 8:12
Derek HoltDerek Holt
53.7k53571
$endgroup$
add a comment |
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1 Answer
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$begingroup$
No of course you cannot deduce that? Why would expect to? (Counterexample: $mathtt{SmallGroup}(16,i)$ for $i=2$ and $4$; or for $i=5$ and $6$.)
answered Jan 20 at 8:12
Derek HoltDerek Holt
53.7k53571
$endgroup$
add a comment |
$begingroup$
No of course you cannot deduce that? Why would expect to? (Counterexample: $mathtt{SmallGroup}(16,i)$ for $i=2$ and $4$; or for $i=5$ and $6$.)
answered Jan 20 at 8:12
Derek HoltDerek Holt
53.7k53571
$endgroup$
add a comment |
$begingroup$
No of course you cannot deduce that? Why would expect to? (Counterexample: $mathtt{SmallGroup}(16,i)$ for $i=2$ and $4$; or for $i=5$ and $6$.)
answered Jan 20 at 8:12
Derek HoltDerek Holt
53.7k53571
$endgroup$
No of course you cannot deduce that? Why would expect to? (Counterexample: $mathtt{SmallGroup}(16,i)$ for $i=2$ and $4$; or for $i=5$ and $6$.)
answered Jan 20 at 8:12
Derek HoltDerek Holt
53.7k53571
answered Jan 20 at 8:12
Derek HoltDerek Holt
53.7k53571
answered Jan 20 at 8:12
Derek HoltDerek Holt
53.7k53571
answered Jan 20 at 8:12
Derek HoltDerek Holt
53.7k53571
53.7k53571
add a comment |
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