Assign a grade of A(correct), C (partially correct), or F (failure) to each. Justifyassignments of grades other than A.(a) Claim. If G is a group with identity e, then G is abelian.Proof. Let a and b be elements of G. Then ba.= e(ba)= (aa1)(ba)= a((a1b)a)= a((b1a)1a)= a(a(b1a)1)= (aa)(b1a)1= (aa)a1b= (aa)(bb1)a1b= a(ab)(b1a1)b= a(ab)(ab)1bT herefore and G is abelian. (b) Claim. If G is a group with elements x, y, and z, and if thenProof. If then implies that so Ifthen the inverse of z exists, and implies andHence in all cases, if then (c) Claim. The set of positive rationals with the operation of multiplicationis a group.Proof. The product of two positive rationals is a positive rational,so is closed under multiplication. Since for every1 is the identity. The inverse of the positive rational a is theb r , 1 r = r = r 1 ++xz = yz, x = y.xz x = y.z= yzz xz = yz(d) Claim. If m is prime, then has no divisors of zero.Proof. Suppose a is a divisor of zero in Thenand there exists in such that Then soThis contradicts the assumption that m is prime. (e) Claim. If m is prime, then has no divisors of zero.Proof. Suppose a is a divisor of zero in Thenand there exists in such that Then som divides ab. Since m is prime, m divides a or m divides b. But since aand b are elements of both are less than m. This is impossible. (f) Claim. For every natural number m, is a group.Proof. We know that is associative with identity element 1.Therefore, is associative with identity element 1. Itremains to show every element has an inverse. ForTherefore, and Therefore, everyelement of has an inverse. (g) Claim. If is a group, then m is prime.Proof. Assume that is a group. Suppose m is notprime. Let where r and s are integers greater than 1 and less thanm. Then Since r has an inverse t inThenThat is, This is impossible, becausez = e, xz = yz xe = ye, x = y. z = e,x = y.xz = yz,ab = ba= ba.= e(ba)= (aa1)(ba)= a((a1b)a)= a((b1a)1a)= a(a(b1a)1)= (aa)(b1a)1= (aa)a1b= (aa)(bb1)a1b= a(ab)(b1a1)b= a(ab)(ab)1bab = aeb{1, 1, i, i}{1, a, b}{1, 1}2x = 4 2x = 3 2x + 3 = 14 + x = 6 x + 7 = 3 3 + x = 1(8, +),6.2 Groups 291positive rational The rationals are associative under multiplicationbecause the reals are associative under multiplication.

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