7. Let o E Sx. a.) Show that the relation, ay if o"(x) = y for some integer n, is an equivalence relation on X. b.) Defi

[answerhappygod]

7 Let O E Sx A Show That The Relation Ay If O X Y For Some Integer N Is An Equivalence Relation On X B Defi 1 (35.62 KiB) Viewed 27 times

7. Let o E Sx. a.) Show that the relation, ay if o"(x) = y for some integer n, is an equivalence relation on X. b.) Define the orbit of a EX with respect to o to be the set Oo(x) = {ye X|x~y}. Compute the orbit of each element in X = {1, 2, 3, 4, 5} with respect to a = (123) (45) Sx. = c.) Show that if Oo(a) nO.(y) # 0, then O.(x) = O. (y). The orbits, with respect to a permutation o, are the equivalence classes for the equivalence relation ~. d.) A subgroup H of Sx is called transitive if for any x, y E X, there exists a y EH such that y(x) = y. Prove that (o) is transitive if and only if Og(x)= X for some x E X.