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

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.

• What do you mean by "the general case", exactly? – Omnomnomnom Jul 20 '18 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 '18 at 12:38
• For an arbitrary polynomial on $x_i$? – Omnomnomnom Jul 20 '18 at 12:39
• Yes , if also polynomial being complicated, for arbitrary polynomial on $x$ – Sandip Agarwal Jul 20 '18 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 '18 at 7:37

The fundamental reason behind this is that you're effectively asking "what happens when permutations act on an arbitrary polynomial?" Well, you can't specify neither the permutations nor the polynomial - this is just intractable (or the answer is simply "anything can happen"). It's like asking "what happens when I evaluate $$P(x)$$ for some $$x$$ and some $$P(x)$$?" Well, anything.
So maybe you'd like to ask "Which polynomials are stabilized by $$H \leq S_n$$?" Or maybe you'd like to ask "Given a specific $$P(x_1,\dots,x_n)$$, is there a better way to find the subgroup of $$S_n$$ that stabilizes it than simply trying out every permutation?"
Otherwise, generally: key ideas would be looking at the representation theory of $$S_n$$, characters of an irreducible representation of $$S_n$$, and if you were interested in subgroups of $$S_n$$, then you would be looking at restricted representations. Sagan's book is a good introduction to the theory.