Modern Algebra (Abstract Algebra) question

(i) Let S3 act on R" by the permutation matrices P(o), where P(o) permutes the basis vectors: P(o) (ei) = es(i). Show that, with this formula, o . (21, X2, 23) = P(o) (x121 +x2e2 + x3e3) = (26-1(1), 26-1(2), 26-1(3) ), and that this is in fact an action on R3. (Note: if you consider instead the more natural looking definition o . (x1, X2, 23) = (X. (1), .(2), o(3) ) then it turns out that T . (o . (x1, X2, X3) ) = (Or) . (x1, 22, 23), with the order reversed.) Find the orbit and the isotropy subgroup of v for v = (1, 2, 3); v = (1, 1, 2); v = (1, 1, 1). (Be careful not to confuse this action with the one from the previous problem!) Verify in all cases that the number of elements of the orbit divides the order of S3. (ii) Let S4 act on R4 by analogy with (i), i.e. o(x1, X2, X3, 24) = (6-1(1), 20-1(2), 26-1(3), No-1(4)). Describe (in a conceptual way, without listing all of the elements) the isotropy subgroup of v for v = (1, 2, 3, 4); v = (1, 1, 2, 3); v =(1,1,1,2); v = (1,1,2,2); 'U = (1,1,1,l). How many elements are there in the orbit for each of the '0 above? (Please don't list all of the elements in the orbits, just describe each orbit and say how many elements it has.) Verify in all cases that the number of elements of the orbit divides the order of S4