Group of order greater than 8 doesn't decompose into a direct product and Sylow 2-subgroup isomorphic...
-1
Is there a group of order greater than 8 that does not decompose into a direct product such that its Sylow 2-subgroup isomorphic quaternion group $Q_8$?
group-theory sylow-theory direct-product
edited Dec 8 at 10:47
Shaun
8,507113580
asked Dec 8 at 10:35
DumbSimon
163
add a comment |
-1
Is there a group of order greater than 8 that does not decompose into a direct product such that its Sylow 2-subgroup isomorphic quaternion group $Q_8$?
group-theory sylow-theory direct-product
edited Dec 8 at 10:47
Shaun
8,507113580
asked Dec 8 at 10:35
DumbSimon
163
You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
– Shaun
Dec 8 at 10:45
add a comment |
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-1
Is there a group of order greater than 8 that does not decompose into a direct product such that its Sylow 2-subgroup isomorphic quaternion group $Q_8$?
group-theory sylow-theory direct-product
edited Dec 8 at 10:47
Shaun
8,507113580
asked Dec 8 at 10:35
DumbSimon
163
Is there a group of order greater than 8 that does not decompose into a direct product such that its Sylow 2-subgroup isomorphic quaternion group $Q_8$?
group-theory sylow-theory direct-product
group-theory sylow-theory direct-product
edited Dec 8 at 10:47
Shaun
8,507113580
asked Dec 8 at 10:35
DumbSimon
163
edited Dec 8 at 10:47
Shaun
8,507113580
asked Dec 8 at 10:35
DumbSimon
163
edited Dec 8 at 10:47
Shaun
8,507113580
edited Dec 8 at 10:47
Shaun
8,507113580
edited Dec 8 at 10:47
Shaun
8,507113580
8,507113580
asked Dec 8 at 10:35
DumbSimon
163
asked Dec 8 at 10:35
DumbSimon
163
asked Dec 8 at 10:35
DumbSimon
163
163
You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
– Shaun
Dec 8 at 10:45
add a comment |
You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
– Shaun
Dec 8 at 10:45
You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
– Shaun
Dec 8 at 10:45
You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
– Shaun
Dec 8 at 10:45
add a comment |
1 Answer
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Hint: $SL(2,3)$ of order $24$.
answered Dec 8 at 14:55
Nicky Hekster
28.2k53456
Thanks for answer. 24 it is order of $S_4$, then 2-sylow subgroup exits. It seems to be isomorphic to $Q_8$. But how to prove it?
– DumbSimon
Dec 8 at 15:37
A Sylow $2$-subgroup of $S_4 $ is isomorphic to $D_4$ the dihedral group of $8$ elements (which is not isomorphic to $Q_8$). More precisely, $langle (24),(1234)rangle$ is a Sylow $2$-subgroup of $S_4$.
– Nicky Hekster
Dec 8 at 18:09
add a comment |
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1 Answer
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Hint: $SL(2,3)$ of order $24$.
answered Dec 8 at 14:55
Nicky Hekster
28.2k53456
Thanks for answer. 24 it is order of $S_4$, then 2-sylow subgroup exits. It seems to be isomorphic to $Q_8$. But how to prove it?
– DumbSimon
Dec 8 at 15:37
A Sylow $2$-subgroup of $S_4 $ is isomorphic to $D_4$ the dihedral group of $8$ elements (which is not isomorphic to $Q_8$). More precisely, $langle (24),(1234)rangle$ is a Sylow $2$-subgroup of $S_4$.
– Nicky Hekster
Dec 8 at 18:09
add a comment |
0
Hint: $SL(2,3)$ of order $24$.
answered Dec 8 at 14:55
Nicky Hekster
28.2k53456
Thanks for answer. 24 it is order of $S_4$, then 2-sylow subgroup exits. It seems to be isomorphic to $Q_8$. But how to prove it?
– DumbSimon
Dec 8 at 15:37
A Sylow $2$-subgroup of $S_4 $ is isomorphic to $D_4$ the dihedral group of $8$ elements (which is not isomorphic to $Q_8$). More precisely, $langle (24),(1234)rangle$ is a Sylow $2$-subgroup of $S_4$.
– Nicky Hekster
Dec 8 at 18:09
add a comment |
0
0
0
Hint: $SL(2,3)$ of order $24$.
answered Dec 8 at 14:55
Nicky Hekster
28.2k53456
Hint: $SL(2,3)$ of order $24$.
answered Dec 8 at 14:55
Nicky Hekster
28.2k53456
answered Dec 8 at 14:55
Nicky Hekster
28.2k53456
answered Dec 8 at 14:55
Nicky Hekster
28.2k53456
answered Dec 8 at 14:55
Nicky Hekster
28.2k53456
28.2k53456
Thanks for answer. 24 it is order of $S_4$, then 2-sylow subgroup exits. It seems to be isomorphic to $Q_8$. But how to prove it?
– DumbSimon
Dec 8 at 15:37
A Sylow $2$-subgroup of $S_4 $ is isomorphic to $D_4$ the dihedral group of $8$ elements (which is not isomorphic to $Q_8$). More precisely, $langle (24),(1234)rangle$ is a Sylow $2$-subgroup of $S_4$.
– Nicky Hekster
Dec 8 at 18:09
add a comment |
Thanks for answer. 24 it is order of $S_4$, then 2-sylow subgroup exits. It seems to be isomorphic to $Q_8$. But how to prove it?
– DumbSimon
Dec 8 at 15:37
A Sylow $2$-subgroup of $S_4 $ is isomorphic to $D_4$ the dihedral group of $8$ elements (which is not isomorphic to $Q_8$). More precisely, $langle (24),(1234)rangle$ is a Sylow $2$-subgroup of $S_4$.
– Nicky Hekster
Dec 8 at 18:09
Thanks for answer. 24 it is order of $S_4$, then 2-sylow subgroup exits. It seems to be isomorphic to $Q_8$. But how to prove it?
– DumbSimon
Dec 8 at 15:37
Thanks for answer. 24 it is order of $S_4$, then 2-sylow subgroup exits. It seems to be isomorphic to $Q_8$. But how to prove it?
– DumbSimon
Dec 8 at 15:37
A Sylow $2$-subgroup of $S_4 $ is isomorphic to $D_4$ the dihedral group of $8$ elements (which is not isomorphic to $Q_8$). More precisely, $langle (24),(1234)rangle$ is a Sylow $2$-subgroup of $S_4$.
– Nicky Hekster
Dec 8 at 18:09
A Sylow $2$-subgroup of $S_4 $ is isomorphic to $D_4$ the dihedral group of $8$ elements (which is not isomorphic to $Q_8$). More precisely, $langle (24),(1234)rangle$ is a Sylow $2$-subgroup of $S_4$.
– Nicky Hekster
Dec 8 at 18:09
add a comment |
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You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
– Shaun
Dec 8 at 10:45