Group Isomorphism regarding Sylow Subgroups
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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 5 hours ago
Ethan Bolker
40.1k544106
asked 5 hours ago
Subhajit Saha
253113
add a comment |
up vote
3
down vote
favorite
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 5 hours ago
Ethan Bolker
40.1k544106
asked 5 hours ago
Subhajit Saha
253113
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
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 5 hours ago
Ethan Bolker
40.1k544106
asked 5 hours ago
Subhajit Saha
253113
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 5 hours ago
Ethan Bolker
40.1k544106
asked 5 hours ago
Subhajit Saha
253113
edited 5 hours ago
Ethan Bolker
40.1k544106
asked 5 hours ago
Subhajit Saha
253113
edited 5 hours ago
Ethan Bolker
40.1k544106
edited 5 hours ago
Ethan Bolker
40.1k544106
edited 5 hours ago
Ethan Bolker
40.1k544106
40.1k544106
asked 5 hours ago
Subhajit Saha
253113
asked 5 hours ago
Subhajit Saha
253113
asked 5 hours ago
Subhajit Saha
253113
253113
add a comment |
add a comment |
2 Answers
2
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up vote
4
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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 5 hours ago
Matt Samuel
36.3k63464
Ok! But what about those cases if we have distinct syllow subgroups of different order
– Subhajit Saha
4 hours ago
@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
4 hours ago
add a comment |
up vote
2
down vote
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 5 hours ago
Eric Towers
31.4k22265
@MattSamuel D'oh. You are correct.
– Eric Towers
5 hours ago
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
5 hours ago
@SubhajitSaha : All Sylow subgroups cyclic implies metacyclic, which does not imply abelian.
– Eric Towers
3 hours ago
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
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 5 hours ago
Matt Samuel
36.3k63464
Ok! But what about those cases if we have distinct syllow subgroups of different order
– Subhajit Saha
4 hours ago
@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
4 hours ago
add a comment |
up vote
4
down vote
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 5 hours ago
Matt Samuel
36.3k63464
Ok! But what about those cases if we have distinct syllow subgroups of different order
– Subhajit Saha
4 hours ago
@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
4 hours ago
add a comment |
up vote
4
down vote
up vote
4
down vote
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 5 hours ago
Matt Samuel
36.3k63464
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 5 hours ago
Matt Samuel
36.3k63464
answered 5 hours ago
Matt Samuel
36.3k63464
answered 5 hours ago
Matt Samuel
36.3k63464
answered 5 hours ago
Matt Samuel
36.3k63464
36.3k63464
Ok! But what about those cases if we have distinct syllow subgroups of different order
– Subhajit Saha
4 hours ago
@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
4 hours ago
add a comment |
Ok! But what about those cases if we have distinct syllow subgroups of different order
– Subhajit Saha
4 hours ago
@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
4 hours ago
Ok! But what about those cases if we have distinct syllow subgroups of different order
– Subhajit Saha
4 hours ago
Ok! But what about those cases if we have distinct syllow subgroups of different order
– Subhajit Saha
4 hours ago
@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
4 hours ago
@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
4 hours ago
add a comment |
up vote
2
down vote
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 5 hours ago
Eric Towers
31.4k22265
@MattSamuel D'oh. You are correct.
– Eric Towers
5 hours ago
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
5 hours ago
@SubhajitSaha : All Sylow subgroups cyclic implies metacyclic, which does not imply abelian.
– Eric Towers
3 hours ago
add a comment |
up vote
2
down vote
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 5 hours ago
Eric Towers
31.4k22265
@MattSamuel D'oh. You are correct.
– Eric Towers
5 hours ago
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
5 hours ago
@SubhajitSaha : All Sylow subgroups cyclic implies metacyclic, which does not imply abelian.
– Eric Towers
3 hours ago
add a comment |
up vote
2
down vote
up vote
2
down vote
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 5 hours ago
Eric Towers
31.4k22265
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 5 hours ago
Eric Towers
31.4k22265
edited 5 hours ago
answered 5 hours ago
Eric Towers
31.4k22265
answered 5 hours ago
Eric Towers
31.4k22265
answered 5 hours ago
Eric Towers
31.4k22265
31.4k22265
@MattSamuel D'oh. You are correct.
– Eric Towers
5 hours ago
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
5 hours ago
@SubhajitSaha : All Sylow subgroups cyclic implies metacyclic, which does not imply abelian.
– Eric Towers
3 hours ago
add a comment |
@MattSamuel D'oh. You are correct.
– Eric Towers
5 hours ago
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
5 hours ago
@SubhajitSaha : All Sylow subgroups cyclic implies metacyclic, which does not imply abelian.
– Eric Towers
3 hours ago
@MattSamuel D'oh. You are correct.
– Eric Towers
5 hours ago
@MattSamuel D'oh. You are correct.
– Eric Towers
5 hours ago
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
5 hours ago
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
5 hours ago
@SubhajitSaha : All Sylow subgroups cyclic implies metacyclic, which does not imply abelian.
– Eric Towers
3 hours ago
@SubhajitSaha : All Sylow subgroups cyclic implies metacyclic, which does not imply abelian.
– Eric Towers
3 hours ago
add a comment |
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