Dummit -Foote Abstract Algebra Chap.2 sec 2.2 problem 12 (e)

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Dummit-Foote Abstract Algebra
Chap.2 sec 2.2 problem 12 (e)



Exhibit all permutation in $S_4$ that stabilize the element $x_1x_2+x_3x_4$ and prove that they formed a subgroup isomorphic to the dihedral group of order 8.



I can solve this
$(1),(12), (34), (12)(34), (1324),(13)(24),(1423), (14)(23) $



But I can't understand how to handle a general case, for $S_n $.
Please help.
Thanks for reading.




abstract-algebra group-theory symmetric-groups




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  • What do you mean by "the general case", exactly?
    – Omnomnomnom
    Jul 20 at 12:35










  • I want to know if the problem asked for $S_n $ for large n, then is there any general rule for this?
    – Sandip Agarwal
    Jul 20 at 12:38










  • For an arbitrary polynomial on $x_i$?
    – Omnomnomnom
    Jul 20 at 12:39











  • Yes , if also polynomial being complicated, for arbitrary polynomial on $x $
    – Sandip Agarwal
    Jul 20 at 12:41










  • It's going to be gross for general $S_n$. There's a good reason the exercise in Dummit and Foote only deals with $n leq 4$. :P
    – Mike Pierce
    Jul 21 at 7:37















up vote
0
down vote

favorite
1












Dummit-Foote Abstract Algebra
Chap.2 sec 2.2 problem 12 (e)



Exhibit all permutation in $S_4$ that stabilize the element $x_1x_2+x_3x_4$ and prove that they formed a subgroup isomorphic to the dihedral group of order 8.



I can solve this
$(1),(12), (34), (12)(34), (1324),(13)(24),(1423), (14)(23) $



But I can't understand how to handle a general case, for $S_n $.
Please help.
Thanks for reading.




abstract-algebra group-theory symmetric-groups




share|cite|improve this question





















  • What do you mean by "the general case", exactly?
    – Omnomnomnom
    Jul 20 at 12:35










  • I want to know if the problem asked for $S_n $ for large n, then is there any general rule for this?
    – Sandip Agarwal
    Jul 20 at 12:38










  • For an arbitrary polynomial on $x_i$?
    – Omnomnomnom
    Jul 20 at 12:39











  • Yes , if also polynomial being complicated, for arbitrary polynomial on $x $
    – Sandip Agarwal
    Jul 20 at 12:41










  • It's going to be gross for general $S_n$. There's a good reason the exercise in Dummit and Foote only deals with $n leq 4$. :P
    – Mike Pierce
    Jul 21 at 7:37













up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





Dummit-Foote Abstract Algebra
Chap.2 sec 2.2 problem 12 (e)



Exhibit all permutation in $S_4$ that stabilize the element $x_1x_2+x_3x_4$ and prove that they formed a subgroup isomorphic to the dihedral group of order 8.



I can solve this
$(1),(12), (34), (12)(34), (1324),(13)(24),(1423), (14)(23) $



But I can't understand how to handle a general case, for $S_n $.
Please help.
Thanks for reading.




abstract-algebra group-theory symmetric-groups




share|cite|improve this question













Dummit-Foote Abstract Algebra
Chap.2 sec 2.2 problem 12 (e)



Exhibit all permutation in $S_4$ that stabilize the element $x_1x_2+x_3x_4$ and prove that they formed a subgroup isomorphic to the dihedral group of order 8.



I can solve this
$(1),(12), (34), (12)(34), (1324),(13)(24),(1423), (14)(23) $



But I can't understand how to handle a general case, for $S_n $.
Please help.
Thanks for reading.





abstract-algebra group-theory symmetric-groups





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 20 at 13:28









Bernard

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asked Jul 20 at 12:25









Sandip Agarwal

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  • What do you mean by "the general case", exactly?
    – Omnomnomnom
    Jul 20 at 12:35










  • I want to know if the problem asked for $S_n $ for large n, then is there any general rule for this?
    – Sandip Agarwal
    Jul 20 at 12:38










  • For an arbitrary polynomial on $x_i$?
    – Omnomnomnom
    Jul 20 at 12:39











  • Yes , if also polynomial being complicated, for arbitrary polynomial on $x $
    – Sandip Agarwal
    Jul 20 at 12:41










  • It's going to be gross for general $S_n$. There's a good reason the exercise in Dummit and Foote only deals with $n leq 4$. :P
    – Mike Pierce
    Jul 21 at 7:37

















  • What do you mean by "the general case", exactly?
    – Omnomnomnom
    Jul 20 at 12:35










  • I want to know if the problem asked for $S_n $ for large n, then is there any general rule for this?
    – Sandip Agarwal
    Jul 20 at 12:38










  • For an arbitrary polynomial on $x_i$?
    – Omnomnomnom
    Jul 20 at 12:39











  • Yes , if also polynomial being complicated, for arbitrary polynomial on $x $
    – Sandip Agarwal
    Jul 20 at 12:41










  • It's going to be gross for general $S_n$. There's a good reason the exercise in Dummit and Foote only deals with $n leq 4$. :P
    – Mike Pierce
    Jul 21 at 7:37
















What do you mean by "the general case", exactly?
– Omnomnomnom
Jul 20 at 12:35




What do you mean by "the general case", exactly?
– Omnomnomnom
Jul 20 at 12:35












I want to know if the problem asked for $S_n $ for large n, then is there any general rule for this?
– Sandip Agarwal
Jul 20 at 12:38




I want to know if the problem asked for $S_n $ for large n, then is there any general rule for this?
– Sandip Agarwal
Jul 20 at 12:38












For an arbitrary polynomial on $x_i$?
– Omnomnomnom
Jul 20 at 12:39





For an arbitrary polynomial on $x_i$?
– Omnomnomnom
Jul 20 at 12:39













Yes , if also polynomial being complicated, for arbitrary polynomial on $x $
– Sandip Agarwal
Jul 20 at 12:41




Yes , if also polynomial being complicated, for arbitrary polynomial on $x $
– Sandip Agarwal
Jul 20 at 12:41












It's going to be gross for general $S_n$. There's a good reason the exercise in Dummit and Foote only deals with $n leq 4$. :P
– Mike Pierce
Jul 21 at 7:37





It's going to be gross for general $S_n$. There's a good reason the exercise in Dummit and Foote only deals with $n leq 4$. :P
– Mike Pierce
Jul 21 at 7:37
















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