7. Let o = (15)(26 9 7 8) (34), 7= (1 3 5)(29) € Sg and G=(0,7) < Sg. (a) (7 points) Find the composition a OT, expressed in disjoint cycle notation. JOT= (b) (7 points) Is a even or odd? Circle the correct answer. EVEN ODD (c) (7 points) Compute the order of a. |o|= (d) (7 points) Considering the action of G on {1,2,3,4,5,6,7,8,9} coming from the inclusion into S9, list the elements of the orbit of G. G.1= (e) (7 points) With the action of G in part (d), we have stabc(1) has 360 elements (you do not need to prove this). What is the order of G? Hint: Use your answer to part (d), even if you couldn't find G.1. |G|=

Exercise 3.7.8 For every n23, construct a nonabelian group of order ny(n), where y(n) = Z is the Euler phi function of n. Hint: Use Exercise 3.7.7.

Exercise 4.1.1 Prove Proposition 4.1.5: Suppose Gx X → X is an action. Then for all z € X, stabĠ(z) is a subgroup of G and the kernel K of the action is the intersection of all stabilizers

Exercise 4.2.3 Edges e, e' of a tetrahedron T are said to be opposite if they are disjoint (that is, they do not share a vertex). The 6 edges can be partitioned into a set X of three pairs of opposite edges. Prove that G3, the group of symmetries of T, acts on X and the kernel K< G3 is a normal subgroup of order 4. The previous exercise is interesting in that it provides a geometric description of a normal subgroup of G3 S₁ different than A₁. It turns out that for any n 4, the only normal subgroup of S₁ is An (and {(1)} and Sn, of course).

Exercise 4.3.1 Verify the claims in the proof of Theorem 4.3.2. Specifically, prove that (i) a - a (91,..., 9p) = (ga+1,..., ga+p) defines an action of Z, on GP and (ii) X = {(9₁,..., 9p) EGP | 91929p = e} is invariant by Zp and hence the Z, action on GP restricts to an action on X. Hint: for the last part, you need to show that for all a € Zp, 91 9p = e if and only if ga+1 ga+p = e.

Exercise 4.3.4 Let p≥ 2 be a prime and prove that any group of order p² is isomorphic to either Zp² or Zpx Zp. Hint: If there exists an element of order p², then GZp², so assume that this is not the case and deduce that every nonidentity element of G has order p. Then prove that for any two elements g, h E G, either (g) = (h) or (g) n(h) = {e}. Finally, prove that there are two nonidentity elements g, h E G so that (g) n(h) = {e}, and apply Proposition 3.7.1.

Exercise 3 Let G be a group of order 52-72-19-23275. (a) Prove that G contains exactly one subgroup of order 49. Prove furthermore that if N <G with |N| then N is normal. (b) Prove that G/N is isomorphic to either Z19 × Z25 or Z19 × Z5 X Z Suggestion: Similar to the Exercise 2c, exept apply Proposition 3.7.1 instead of 3.7.7. 49 (c) Let Ps and P19 be Sylow 5- and 19-subgroups of G, respectively. Prove that NP5 and NP19 are both subgroups of G and that NPN X P5, and NP19 N x P19-

Let A CR, and let x Є R. We say that the point x is far away from the set A if and only if:Ed >0: No element of A belongs to the set [x − d, x].

Exercise 4 Let G be a group of order 2³ - 7 = 56. Prove that G contains at least one normal subgroup. Suggestion: Prove that G contains either a normal Sylow 2-subgroup or a normal Sylow 7-subgroup. Do this by assuming otherwise and counting elements. Be careful with intersections when doing your count.

3.3.3