Elements in The Orbit and Symmetric Group Moldern Algebra Problems

Moldern algebra problem. need a pdf or png after you finish it. you have up to 8 eight hours to finish it.

Problem 1. Consider the symmetric group Sr.
a) Let a = (a1 Q2 -.-as) e Srbe an s-cycle. If B e Sn, show that
Baß ‘ = (B(Q1) B(a2) …B(as)).
In particular, Baß ‘ is an s-cycle.
b) Show that a,B e Shave the same cycle type if and only if there is a y E Son such that
α= γβγ !.
(Hint: the backwards direction should follow quickly from part a. For the backwards direction,
try showing that (1 3 5)(2 4) and (3 2 4)(15) are conjugates as a warm up to the general case)
Problem 4. Let A = P({1,2,3,4}) denote the power set of {1,2,3,4}.
a) Prove that .: S4A A given by a. S = a[S] defines a group action of S4 on A.
b) Determine the distinct orbits of the group action from part a. Verify that they form a
partition of A.
c) For each of the orbits found in part b, compute the stabilizer of one of the elements in the
orbit (just choose whichever one you like). Check that the conclusion of the orbit-stabilizer
theorem holds.