Problem 20. Is R a subgroup of S? If so, is it a normal subgroup? Is...
Problem 7. (20 pts) For each of the ten sentences below, justify whether they are true or false. If true, you must provide a proof, if false you must provide a counter-example. (a) The linear map ƒ : R² → R² defined by (:)-()) is an isometry. (b) Any linear map f : R² → R² of the form (;)-( :)(;). with a + 0, must be an isometry. (c) The composition fog : R² → R? of two isometries...
(5 points) Recall the Definition: A subgroup H of G is called a normal subgroup of G if gH = Hg for all g E G. If so, we write H G. Mark each of the following true T or false F (using the CAPITAL LETTER T or F. Recall that if a statment is not necessarily ALWAYS true, then it is false. - T ח 1. Every subgroup of (Zn, e) is normal. 2. The cyclic group (f) is...
Problem 6. (20 pts.) Let R = R\{0, 1,2) = {r€R ]r#0,1,2} be the set of all real numbers except 0,1,2. Let G be a subgroup of the group of bijective functions Describe all elements of G and construct the Cayley diagram for G. What familiar group is G isomorphic to (construct the isomorphism erplicitly)? R, PR, generated by f(r) 2-r and g(z) 2/ . on Problem 6. (20 pts.) Let R = R\{0, 1,2) = {r€R ]r#0,1,2} be the...
(a) Let R be a commutative ring. Given a finite subset {ai, a2, , an} of R, con- sider the set {rial + r202 + . . . + rnan I ri, r2, . . . , rn є R), which we denote by 〈a1, a2 , . . . , Prove that 〈a1, a2, . . . , an〉 įs an ideal of R. (If an ideal 1 = 〈a1, аг, . . . , an) for some a,...
Let F = R. Let f = t3 – ajta – azt Az E R[t]. Show: (a) The discriminant A = -4aſaz + a až – 18a1a2a3 + 4a3 – 27az. (b) f has multiple roots if and only if A = 0. (c) f has three distinct real roots if and only if A >0. (d) f has one real root and two non-real roots if and only if A < 0.
2. Assume the group G acts on the set S. For E S, define Then G is a subgroup of G , which is called the stabilizer of r. The set is called the orbit of r (a) Consider the map ф' G S, defined by фг (g) :-9-x. Prove that there is one map (and only one) : G/G, S such that Vz ยู่'z q (where q: G -G/G, is the quotient map). (b) Prove that is injective. (Hint:...
Let S be the solid of revolution obtained by revolving the region R of the z y plane about the line z 4where R is the region defined by the curves -6 andy-6- We wish to compute the volume of S by using the method of cylindrical shells a) Determine the smallest x-coordinate 1 and the largest x-coordinate r2 of the points in this region b) Let x be a real number in the interval |1,2 We consider the thin...
Please do all. Thanks. Instructions. Five problems on two pages, ten points each. Throughout, let R be a commutative ring with 1 Definition: For a, beR we say that a divides b (notation: a | b) if there is some z E R with az b Definition: For a E R we say that a is an unit if a | 1. 3. Prove that if a is an unit then for any ceR,a| c. 4. Prove that if a...
Problem 2. Let be the quarter torus with outward normal. Use the parameterization r(u, v) = (4 + 2 cos(v)) cos(u)i + (4 + 2 cos(u)) sin(u)j + 2 sin(v)k, for 0 Susand 0 <0527 (a) Find a parameterization for each of the curves forming the boundary of E. Make sure the orientation of the curves match the orientation induced by S. (b) Let F(x, y, z) = xyi+yzj+rzk. Evaluate S/.( VF) ds.
PROBLEM e Definition: A GROUP is a set S paired with an operation *, denoted <S,*> satisfying the four properties; G0: CLOSURE - For any a, b in S, a * b in S G1: ASSOCIATIVITY - For all a, b, c in S, (a * b) * c = a * (b * c) G2: IDENITY - There exists an element e in S such that a * e = e = b * a, for all a in...