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College Algebra Exam Review 233

# College Algebra Exam Review 233 - HA is determined by.h 1 a...

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5.1. GROUP ACTIONS ON SETS 243 5.1.5. Verify that any group G acts on the set X of its subgroups by c g .H/ D gHg 1 . Compute the example of S 3 acting by conjugation of the set X of (six) subgroups of S 3 . Verify that there are four orbits, three of which consist of a single subgroup, and one of which contains three subgroups. 5.1.6. Let G act on X , and let x 2 X . Verify that Stab .x/ is a subgroup of G . Verify that if x and y are in the same orbit, then the subgroups Stab .x/ and Stab .y/ are conjugate subgroups. 5.1.7. Let H D f e; .1; 2/ g S 3 . Find the orbit of H under conjugation by G , the stabilizer of H in G , and the family of left cosets of the stabilizer in G , and verify explicitly the bijection between left cosets of the stabilizer and conjugates of H . 5.1.8. Show that N G .aHa 1 / D aN G .H/a 1 for a 2 G . 5.1.9. Show that the transitive subgroups of S 3 are exactly S 3 and A 3 . 5.1.10. Suppose A is a subgroup of N G .H/ . Show that AH is a subgroup of N G .H/ , AH D HA , and j AH j D j A j j H j j A \ H j : Hint: H is normal in N G .H/ . 5.1.11. Let A be a subgroup of N G .H/ . Show that there is a homomor- phism ˛ W A ! Aut .H/ (denoted a 7! ˛ a ) such that ah D ˛ a .h/a for all a 2 A and h 2 H . The product in
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Unformatted text preview: HA is determined by .h 1 a 1 /.h 2 a 2 / D h 1 ˛ a 1 .h 2 /a 1 a 2 : 5.1.12. Count the number of ways to arrange four red beads, three blue beads, and two yellow beads on a straight wire. 5.1.13. How may elements are there in S 8 of cycle structure 4 2 ? 5.1.14. How may elements are there in S 8 of cycle structure 2 4 ? 5.1.15. How may elements are there in S 12 of cycle structure 3 2 2 3 ? 5.1.16. How may elements are there in S 26 of cycle structure 4 3 3 2 2 4 ? 5.1.17. Let r 1 > r 2 > ::: > r s ³ 1 and let m i 2 N for 1 ´ i ´ s , such that P i m i r i D n . How many elements does S n have with m 1 cycles of length r 1 , m 2 cycles of length r 2 , and so on? 5.1.18. Verify that the formula ± f a 1 ;a 2 ;:::;a k g D f ±.a 1 /;±.a 2 /;:::;±.a k / g does indeed deﬁne an action of S n on the set X of k-element subsets of f 1;2;:::;n g . (See Example 5.1.19 .)...
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