Group Isomorphism regarding Sylow Subgroups
Suppose I have given two groups say, $G_1,G_2$ such that they have same order.I'm assuming they are non commutative.Then their Syllow subgroups has same order clearly.If I'm given that the number of Syllow subgroups of these are also same then "are $G_1,G_2$ isomorphic"? I have always find this statement as true considering lower order groups but can't proved it. Is it true or there are some counterexamples too! Thanks for reading.
abstract-algebra group-isomorphism sylow-theory
edited Dec 5 at 19:17
Ethan Bolker
41.3k547108
asked Dec 5 at 19:06
Subhajit Saha
257113
add a comment |
Suppose I have given two groups say, $G_1,G_2$ such that they have same order.I'm assuming they are non commutative.Then their Syllow subgroups has same order clearly.If I'm given that the number of Syllow subgroups of these are also same then "are $G_1,G_2$ isomorphic"? I have always find this statement as true considering lower order groups but can't proved it. Is it true or there are some counterexamples too! Thanks for reading.
abstract-algebra group-isomorphism sylow-theory
edited Dec 5 at 19:17
Ethan Bolker
41.3k547108
asked Dec 5 at 19:06
Subhajit Saha
257113
add a comment |
Suppose I have given two groups say, $G_1,G_2$ such that they have same order.I'm assuming they are non commutative.Then their Syllow subgroups has same order clearly.If I'm given that the number of Syllow subgroups of these are also same then "are $G_1,G_2$ isomorphic"? I have always find this statement as true considering lower order groups but can't proved it. Is it true or there are some counterexamples too! Thanks for reading.
abstract-algebra group-isomorphism sylow-theory
edited Dec 5 at 19:17
Ethan Bolker
41.3k547108
asked Dec 5 at 19:06
Subhajit Saha
257113
Suppose I have given two groups say, $G_1,G_2$ such that they have same order.I'm assuming they are non commutative.Then their Syllow subgroups has same order clearly.If I'm given that the number of Syllow subgroups of these are also same then "are $G_1,G_2$ isomorphic"? I have always find this statement as true considering lower order groups but can't proved it. Is it true or there are some counterexamples too! Thanks for reading.
abstract-algebra group-isomorphism sylow-theory
abstract-algebra group-isomorphism sylow-theory
edited Dec 5 at 19:17
Ethan Bolker
41.3k547108
asked Dec 5 at 19:06
Subhajit Saha
257113
edited Dec 5 at 19:17
Ethan Bolker
41.3k547108
asked Dec 5 at 19:06
Subhajit Saha
257113
edited Dec 5 at 19:17
Ethan Bolker
41.3k547108
edited Dec 5 at 19:17
Ethan Bolker
41.3k547108
edited Dec 5 at 19:17
Ethan Bolker
41.3k547108
41.3k547108
asked Dec 5 at 19:06
Subhajit Saha
257113
asked Dec 5 at 19:06
Subhajit Saha
257113
asked Dec 5 at 19:06
Subhajit Saha
257113
257113
add a comment |
add a comment |
2 Answers
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active
oldest
votes
This is easily seen to fail for abelian groups, since all abelian groups of a given order have the same number of Sylow subgroups. For a nonabelian example, consider two distinct nonabelian groups of order $p^n$ for some prime $p$ and integer $n$.
answered Dec 5 at 19:18
Matt Samuel
37.1k63465
Ok! But what about those cases if we have distinct syllow subgroups of different order
– Subhajit Saha
Dec 5 at 19:55
@Sub I don't have an example, but I'm $100%$ certain the numbers of Sylow subgroups doesn't classify the group, abelian or not. It would make the group isomorphism problem easy, which it isn't.
– Matt Samuel
Dec 5 at 20:14
add a comment |
Groups with Identical Subgroup Lattices in All Powers shows there are many, many examples even when the Sylow subgroups are required to be cyclic.
answered Dec 5 at 19:16
Eric Towers
31.8k22265
@MattSamuel D'oh. You are correct.
– Eric Towers
Dec 5 at 19:20
Sir,I said both of them as non Abelian , further you have done a mistake saying $S_3$ has unique $2-$ Syllow subgroups.
– Subhajit Saha
Dec 5 at 19:25
@SubhajitSaha : All Sylow subgroups cyclic implies metacyclic, which does not imply abelian.
– Eric Towers
Dec 5 at 21:43
add a comment |
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2 Answers
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2 Answers
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This is easily seen to fail for abelian groups, since all abelian groups of a given order have the same number of Sylow subgroups. For a nonabelian example, consider two distinct nonabelian groups of order $p^n$ for some prime $p$ and integer $n$.
answered Dec 5 at 19:18
Matt Samuel
37.1k63465
Ok! But what about those cases if we have distinct syllow subgroups of different order
– Subhajit Saha
Dec 5 at 19:55
@Sub I don't have an example, but I'm $100%$ certain the numbers of Sylow subgroups doesn't classify the group, abelian or not. It would make the group isomorphism problem easy, which it isn't.
– Matt Samuel
Dec 5 at 20:14
add a comment |
This is easily seen to fail for abelian groups, since all abelian groups of a given order have the same number of Sylow subgroups. For a nonabelian example, consider two distinct nonabelian groups of order $p^n$ for some prime $p$ and integer $n$.
answered Dec 5 at 19:18
Matt Samuel
37.1k63465
Ok! But what about those cases if we have distinct syllow subgroups of different order
– Subhajit Saha
Dec 5 at 19:55
@Sub I don't have an example, but I'm $100%$ certain the numbers of Sylow subgroups doesn't classify the group, abelian or not. It would make the group isomorphism problem easy, which it isn't.
– Matt Samuel
Dec 5 at 20:14
add a comment |
This is easily seen to fail for abelian groups, since all abelian groups of a given order have the same number of Sylow subgroups. For a nonabelian example, consider two distinct nonabelian groups of order $p^n$ for some prime $p$ and integer $n$.
answered Dec 5 at 19:18
Matt Samuel
37.1k63465
This is easily seen to fail for abelian groups, since all abelian groups of a given order have the same number of Sylow subgroups. For a nonabelian example, consider two distinct nonabelian groups of order $p^n$ for some prime $p$ and integer $n$.
answered Dec 5 at 19:18
Matt Samuel
37.1k63465
answered Dec 5 at 19:18
Matt Samuel
37.1k63465
answered Dec 5 at 19:18
Matt Samuel
37.1k63465
answered Dec 5 at 19:18
Matt Samuel
37.1k63465
37.1k63465
Ok! But what about those cases if we have distinct syllow subgroups of different order
– Subhajit Saha
Dec 5 at 19:55
@Sub I don't have an example, but I'm $100%$ certain the numbers of Sylow subgroups doesn't classify the group, abelian or not. It would make the group isomorphism problem easy, which it isn't.
– Matt Samuel
Dec 5 at 20:14
add a comment |
Ok! But what about those cases if we have distinct syllow subgroups of different order
– Subhajit Saha
Dec 5 at 19:55
@Sub I don't have an example, but I'm $100%$ certain the numbers of Sylow subgroups doesn't classify the group, abelian or not. It would make the group isomorphism problem easy, which it isn't.
– Matt Samuel
Dec 5 at 20:14
Ok! But what about those cases if we have distinct syllow subgroups of different order
– Subhajit Saha
Dec 5 at 19:55
Ok! But what about those cases if we have distinct syllow subgroups of different order
– Subhajit Saha
Dec 5 at 19:55
@Sub I don't have an example, but I'm $100%$ certain the numbers of Sylow subgroups doesn't classify the group, abelian or not. It would make the group isomorphism problem easy, which it isn't.
– Matt Samuel
Dec 5 at 20:14
@Sub I don't have an example, but I'm $100%$ certain the numbers of Sylow subgroups doesn't classify the group, abelian or not. It would make the group isomorphism problem easy, which it isn't.
– Matt Samuel
Dec 5 at 20:14
add a comment |
Groups with Identical Subgroup Lattices in All Powers shows there are many, many examples even when the Sylow subgroups are required to be cyclic.
answered Dec 5 at 19:16
Eric Towers
31.8k22265
@MattSamuel D'oh. You are correct.
– Eric Towers
Dec 5 at 19:20
Sir,I said both of them as non Abelian , further you have done a mistake saying $S_3$ has unique $2-$ Syllow subgroups.
– Subhajit Saha
Dec 5 at 19:25
@SubhajitSaha : All Sylow subgroups cyclic implies metacyclic, which does not imply abelian.
– Eric Towers
Dec 5 at 21:43
add a comment |
Groups with Identical Subgroup Lattices in All Powers shows there are many, many examples even when the Sylow subgroups are required to be cyclic.
answered Dec 5 at 19:16
Eric Towers
31.8k22265
@MattSamuel D'oh. You are correct.
– Eric Towers
Dec 5 at 19:20
Sir,I said both of them as non Abelian , further you have done a mistake saying $S_3$ has unique $2-$ Syllow subgroups.
– Subhajit Saha
Dec 5 at 19:25
@SubhajitSaha : All Sylow subgroups cyclic implies metacyclic, which does not imply abelian.
– Eric Towers
Dec 5 at 21:43
add a comment |
Groups with Identical Subgroup Lattices in All Powers shows there are many, many examples even when the Sylow subgroups are required to be cyclic.
answered Dec 5 at 19:16
Eric Towers
31.8k22265
Groups with Identical Subgroup Lattices in All Powers shows there are many, many examples even when the Sylow subgroups are required to be cyclic.
answered Dec 5 at 19:16
Eric Towers
31.8k22265
edited Dec 5 at 19:20
answered Dec 5 at 19:16
Eric Towers
31.8k22265
answered Dec 5 at 19:16
Eric Towers
31.8k22265
answered Dec 5 at 19:16
Eric Towers
31.8k22265
31.8k22265
@MattSamuel D'oh. You are correct.
– Eric Towers
Dec 5 at 19:20
Sir,I said both of them as non Abelian , further you have done a mistake saying $S_3$ has unique $2-$ Syllow subgroups.
– Subhajit Saha
Dec 5 at 19:25
@SubhajitSaha : All Sylow subgroups cyclic implies metacyclic, which does not imply abelian.
– Eric Towers
Dec 5 at 21:43
add a comment |
@MattSamuel D'oh. You are correct.
– Eric Towers
Dec 5 at 19:20
Sir,I said both of them as non Abelian , further you have done a mistake saying $S_3$ has unique $2-$ Syllow subgroups.
– Subhajit Saha
Dec 5 at 19:25
@SubhajitSaha : All Sylow subgroups cyclic implies metacyclic, which does not imply abelian.
– Eric Towers
Dec 5 at 21:43
@MattSamuel D'oh. You are correct.
– Eric Towers
Dec 5 at 19:20
@MattSamuel D'oh. You are correct.
– Eric Towers
Dec 5 at 19:20
Sir,I said both of them as non Abelian , further you have done a mistake saying $S_3$ has unique $2-$ Syllow subgroups.
– Subhajit Saha
Dec 5 at 19:25
Sir,I said both of them as non Abelian , further you have done a mistake saying $S_3$ has unique $2-$ Syllow subgroups.
– Subhajit Saha
Dec 5 at 19:25
@SubhajitSaha : All Sylow subgroups cyclic implies metacyclic, which does not imply abelian.
– Eric Towers
Dec 5 at 21:43
@SubhajitSaha : All Sylow subgroups cyclic implies metacyclic, which does not imply abelian.
– Eric Towers
Dec 5 at 21:43
add a comment |
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