Group Theory Sylow Subgroup [closed]
$begingroup$
What's an example of a group $G$ and an integer $n$ dividing $|G|$ with
$0 < n < |G|$ such that $G$ has no subgroup of order $n$.
group-theory sylow-theory
$endgroup$
closed as off-topic by ml0105, Randall, Saad, Dando18, Lord Shark the Unknown Dec 10 '18 at 3:55
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – ml0105, Randall, Saad, Dando18, Lord Shark the Unknown
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
What's an example of a group $G$ and an integer $n$ dividing $|G|$ with
$0 < n < |G|$ such that $G$ has no subgroup of order $n$.
group-theory sylow-theory
$endgroup$
closed as off-topic by ml0105, Randall, Saad, Dando18, Lord Shark the Unknown Dec 10 '18 at 3:55
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – ml0105, Randall, Saad, Dando18, Lord Shark the Unknown
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Please also consider adding what you have tried in solving the problem. In MSE bare problem statement questions (without attempt from the asker) are generally frowned upon. So please consider adding them.
$endgroup$
– user 170039
Dec 10 '18 at 3:50
$begingroup$
I will. Thanks for letting me know what MSE is like.
$endgroup$
– user624358
Dec 10 '18 at 6:31
add a comment |
$begingroup$
What's an example of a group $G$ and an integer $n$ dividing $|G|$ with
$0 < n < |G|$ such that $G$ has no subgroup of order $n$.
group-theory sylow-theory
$endgroup$
What's an example of a group $G$ and an integer $n$ dividing $|G|$ with
$0 < n < |G|$ such that $G$ has no subgroup of order $n$.
group-theory sylow-theory
group-theory sylow-theory
edited Dec 10 '18 at 2:28
asked Dec 10 '18 at 0:23
user624358
closed as off-topic by ml0105, Randall, Saad, Dando18, Lord Shark the Unknown Dec 10 '18 at 3:55
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – ml0105, Randall, Saad, Dando18, Lord Shark the Unknown
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by ml0105, Randall, Saad, Dando18, Lord Shark the Unknown Dec 10 '18 at 3:55
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – ml0105, Randall, Saad, Dando18, Lord Shark the Unknown
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Please also consider adding what you have tried in solving the problem. In MSE bare problem statement questions (without attempt from the asker) are generally frowned upon. So please consider adding them.
$endgroup$
– user 170039
Dec 10 '18 at 3:50
$begingroup$
I will. Thanks for letting me know what MSE is like.
$endgroup$
– user624358
Dec 10 '18 at 6:31
add a comment |
$begingroup$
Please also consider adding what you have tried in solving the problem. In MSE bare problem statement questions (without attempt from the asker) are generally frowned upon. So please consider adding them.
$endgroup$
– user 170039
Dec 10 '18 at 3:50
$begingroup$
I will. Thanks for letting me know what MSE is like.
$endgroup$
– user624358
Dec 10 '18 at 6:31
$begingroup$
Please also consider adding what you have tried in solving the problem. In MSE bare problem statement questions (without attempt from the asker) are generally frowned upon. So please consider adding them.
$endgroup$
– user 170039
Dec 10 '18 at 3:50
$begingroup$
Please also consider adding what you have tried in solving the problem. In MSE bare problem statement questions (without attempt from the asker) are generally frowned upon. So please consider adding them.
$endgroup$
– user 170039
Dec 10 '18 at 3:50
$begingroup$
I will. Thanks for letting me know what MSE is like.
$endgroup$
– user624358
Dec 10 '18 at 6:31
$begingroup$
I will. Thanks for letting me know what MSE is like.
$endgroup$
– user624358
Dec 10 '18 at 6:31
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
In $A_4$,with $|A_4|=12$ are there subgroup of order $6$?
answered Dec 10 '18 at 0:33
nafhgoodnafhgood
1,805422
$endgroup$
add a comment |
$begingroup$
Suppose $|G|$ is simple non-cyclic, then $G$ is non-abelian (Use Cauchy's Theorem), then $G$ has even order (Apply Feit–Thompson theorem), hence $G$ has no subgroup of order $|G|/2$. So if $G$ is simple non-cyclic, then $|G|=2n$ and $G$ has no subgroup of order $n.$ In particular the monster group has no subgroup of order $$404008712397256437943229952480855378502877184000000000.$$
answered Dec 10 '18 at 0:38
MelodyMelody
80012
$endgroup$
2
$begingroup$
Well, to use the F-T theorem in such an elementary question is killing a fly not with a canyon but with a thermonuclear bomb...To mention the monster group only adds radiation...
$endgroup$
– DonAntonio
Dec 10 '18 at 0:41
$begingroup$
You're right. My first inclination was just to mention $A_5,$ but then I thought why not apply it to a much larger class of groups? Even if it's totally unnecessary.
$endgroup$
– Melody
Dec 10 '18 at 0:44
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
In $A_4$,with $|A_4|=12$ are there subgroup of order $6$?
answered Dec 10 '18 at 0:33
nafhgoodnafhgood
1,805422
$endgroup$
add a comment |
$begingroup$
In $A_4$,with $|A_4|=12$ are there subgroup of order $6$?
answered Dec 10 '18 at 0:33
nafhgoodnafhgood
1,805422
$endgroup$
add a comment |
$begingroup$
In $A_4$,with $|A_4|=12$ are there subgroup of order $6$?
answered Dec 10 '18 at 0:33
nafhgoodnafhgood
1,805422
$endgroup$
In $A_4$,with $|A_4|=12$ are there subgroup of order $6$?
answered Dec 10 '18 at 0:33
nafhgoodnafhgood
1,805422
answered Dec 10 '18 at 0:33
nafhgoodnafhgood
1,805422
answered Dec 10 '18 at 0:33
nafhgoodnafhgood
1,805422
answered Dec 10 '18 at 0:33
nafhgoodnafhgood
1,805422
1,805422
add a comment |
add a comment |
$begingroup$
Suppose $|G|$ is simple non-cyclic, then $G$ is non-abelian (Use Cauchy's Theorem), then $G$ has even order (Apply Feit–Thompson theorem), hence $G$ has no subgroup of order $|G|/2$. So if $G$ is simple non-cyclic, then $|G|=2n$ and $G$ has no subgroup of order $n.$ In particular the monster group has no subgroup of order $$404008712397256437943229952480855378502877184000000000.$$
answered Dec 10 '18 at 0:38
MelodyMelody
80012
$endgroup$
2
$begingroup$
Well, to use the F-T theorem in such an elementary question is killing a fly not with a canyon but with a thermonuclear bomb...To mention the monster group only adds radiation...
$endgroup$
– DonAntonio
Dec 10 '18 at 0:41
$begingroup$
You're right. My first inclination was just to mention $A_5,$ but then I thought why not apply it to a much larger class of groups? Even if it's totally unnecessary.
$endgroup$
– Melody
Dec 10 '18 at 0:44
add a comment |
$begingroup$
Suppose $|G|$ is simple non-cyclic, then $G$ is non-abelian (Use Cauchy's Theorem), then $G$ has even order (Apply Feit–Thompson theorem), hence $G$ has no subgroup of order $|G|/2$. So if $G$ is simple non-cyclic, then $|G|=2n$ and $G$ has no subgroup of order $n.$ In particular the monster group has no subgroup of order $$404008712397256437943229952480855378502877184000000000.$$
answered Dec 10 '18 at 0:38
MelodyMelody
80012
$endgroup$
2
$begingroup$
Well, to use the F-T theorem in such an elementary question is killing a fly not with a canyon but with a thermonuclear bomb...To mention the monster group only adds radiation...
$endgroup$
– DonAntonio
Dec 10 '18 at 0:41
$begingroup$
You're right. My first inclination was just to mention $A_5,$ but then I thought why not apply it to a much larger class of groups? Even if it's totally unnecessary.
$endgroup$
– Melody
Dec 10 '18 at 0:44
add a comment |
$begingroup$
Suppose $|G|$ is simple non-cyclic, then $G$ is non-abelian (Use Cauchy's Theorem), then $G$ has even order (Apply Feit–Thompson theorem), hence $G$ has no subgroup of order $|G|/2$. So if $G$ is simple non-cyclic, then $|G|=2n$ and $G$ has no subgroup of order $n.$ In particular the monster group has no subgroup of order $$404008712397256437943229952480855378502877184000000000.$$
answered Dec 10 '18 at 0:38
MelodyMelody
80012
$endgroup$
Suppose $|G|$ is simple non-cyclic, then $G$ is non-abelian (Use Cauchy's Theorem), then $G$ has even order (Apply Feit–Thompson theorem), hence $G$ has no subgroup of order $|G|/2$. So if $G$ is simple non-cyclic, then $|G|=2n$ and $G$ has no subgroup of order $n.$ In particular the monster group has no subgroup of order $$404008712397256437943229952480855378502877184000000000.$$
answered Dec 10 '18 at 0:38
MelodyMelody
80012
answered Dec 10 '18 at 0:38
MelodyMelody
80012
answered Dec 10 '18 at 0:38
MelodyMelody
80012
answered Dec 10 '18 at 0:38
MelodyMelody
80012
80012
2
$begingroup$
Well, to use the F-T theorem in such an elementary question is killing a fly not with a canyon but with a thermonuclear bomb...To mention the monster group only adds radiation...
$endgroup$
– DonAntonio
Dec 10 '18 at 0:41
$begingroup$
You're right. My first inclination was just to mention $A_5,$ but then I thought why not apply it to a much larger class of groups? Even if it's totally unnecessary.
$endgroup$
– Melody
Dec 10 '18 at 0:44
add a comment |
2
$begingroup$
Well, to use the F-T theorem in such an elementary question is killing a fly not with a canyon but with a thermonuclear bomb...To mention the monster group only adds radiation...
$endgroup$
– DonAntonio
Dec 10 '18 at 0:41
$begingroup$
You're right. My first inclination was just to mention $A_5,$ but then I thought why not apply it to a much larger class of groups? Even if it's totally unnecessary.
$endgroup$
– Melody
Dec 10 '18 at 0:44
2
2
$begingroup$
Well, to use the F-T theorem in such an elementary question is killing a fly not with a canyon but with a thermonuclear bomb...To mention the monster group only adds radiation...
$endgroup$
– DonAntonio
Dec 10 '18 at 0:41
$begingroup$
Well, to use the F-T theorem in such an elementary question is killing a fly not with a canyon but with a thermonuclear bomb...To mention the monster group only adds radiation...
$endgroup$
– DonAntonio
Dec 10 '18 at 0:41
$begingroup$
You're right. My first inclination was just to mention $A_5,$ but then I thought why not apply it to a much larger class of groups? Even if it's totally unnecessary.
$endgroup$
– Melody
Dec 10 '18 at 0:44
$begingroup$
You're right. My first inclination was just to mention $A_5,$ but then I thought why not apply it to a much larger class of groups? Even if it's totally unnecessary.
$endgroup$
– Melody
Dec 10 '18 at 0:44
add a comment |
$begingroup$
Please also consider adding what you have tried in solving the problem. In MSE bare problem statement questions (without attempt from the asker) are generally frowned upon. So please consider adding them.
$endgroup$
– user 170039
Dec 10 '18 at 3:50
$begingroup$
I will. Thanks for letting me know what MSE is like.
$endgroup$
– user624358
Dec 10 '18 at 6:31