I'm going to have to post the full form of this question as I doubt the terminology is universal. So here it goes!

Okay, after that mouthful I am going to sketch the method I used to answer the question. I found the stabilizer, then calculated the order of each element. I did this also for each element of D_8, then constructed a bijective function between the two groups. It was a bit tedious, but I then verified that this was also an homomorphism. So my bijection is an isomorphism between the normalizer and D_8.Let R be the set of all polynomials with integer coefficients in the independent variables x_1, x_2, x_3, x_4, ie. the members of R are finite sums of elements of the form where a is any integer and r_1, ..., r_4 are nonnegative integers. Each gives a permutation of {x_1, ..., x_4} by defining . This may be extended to a map from R to R by defining for all .

Exhibit all permutations in S_4 that stabilize the element and prove that they form a subgroup isomorphic to the dihedral group D_8.

My question is: Is it possible to find the isomorphism more generally, rather than constructing the bijection?

-Dan