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We consider the operation of the symmetric group S4 on the set R[x,y,z,a] through permutation of...

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Advance Math

We consider the operation of the symmetric group S4 onthe set R[x,y,z,a] through permutation of an unknown integer.
a) Calculate the length of the orbit of polynomial x2+y2+z+a. Howmany permutations leave this polynomial unchanged?
b) Is a polynomial of length 5 under this operation possible?
c) Show the existence of polynomials with orbit length 12 and4.

Answer & Explanation Solved by verified expert
4.3 Ratings (616 Votes)
Orbit Stabilizer Theorem Suppose that G is a finitegroup acting on a set X Then for any x in X Orbitx G StabilizerxUp to isomorphism we can consider S4 to be equal tothe symmetric group on xyzaS4 operates on Rxyza by the group operationfxyrafxyzafor all in S4 and f in RxyzaaLet us compute the cardinality of the stabilizer of thepolynomial gxyza x2 y2 z aObserve that if is in the stabilizer of g in S4 then g g and    See Answer
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