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 3.6.12 Prove the First Isomorphism Theorem of Rings: If o: R→S is a surjective ring homomorphism, then there exists a unique ring isomorphism

Exercise 3.7.2 Suppose G is a finite group, H, KG are normal subgroups, ged(|H|, |K|) = 1, and |G| = |H||K|. Prove that G H x K.

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.

2. (7 points) Find god(216,315) and express it in the form 216n +315m for integers n, m. ged(216, 315) =

9. Let p(x) = 1+2+1²+1³ +r¹ € Q[r]. This polynomial is irreducible (you do not need to prove this). Consider the field obtained as the quotient K = Q[r]/I, where I = {fp|ƒ €Q[r]}. Let a = 1 + 1 € K, which is a root of p(x) in K, and recall that every element of K can be expressed in the form a+ba+co²+do²³, with a, b, c, d e Q. (a) (8 points) What is the degree of the extension, [K: Q? [K: Q = (b) (8 points) Compute the product of 1+a and 2-30³, and express it in the form a +bx+ co²+ da, with a, b, c, d € Q. (1 + a)(2-30³) = (c) (8 points) Since K is a field, and a #0, it follows that a is a unit. Find the multiplicative inverse ¹ and express it in the form a+ba+ca²+da³, with a, b, c, d e Q.

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.