MATH 113 Midterm: MATH113 Midterm 2015 Fall

These questions are asking about subrgoups, groups, isormorphic . actually, after 2day, I got a midterm for abstract math test, and I got a sample of midterm test, so I hope to get a solution of all these question. I think it is a lot, but I hope that please show me the answer and I want to know why these answer is this. The reason why these problems are bunch of questions, so I will give big pts for this questions. thanks. you guys can upload by the picture. Write the denition of group. Give an example of a group of order 2011. Give an example of a non-abelian group with 12 elements. Let (G,*) and (H . ) be groups. Dene what it means for G and H to be isomorphic. Assume that G has exactly two solutions to the equation x^2 = e. Prove, directly from the denition in (1), that if G and H are iso- morphic, then H has exactly two solutions to the equation x^2 = e. Prove or disprove: (R, +) is isomorphic (R+, . ). Here R+ = {x is subset of R : x>0}. Prove or disprove: (R, +) is isomorphic (R*, . ). Here R* = {x is subset of R : x is not 0} Let G be the group Z_20, with addition modulo 20. Draw the subgroup diagram of G, giving a generator for each subgroup Find the subgroup H = of G. Find a subgroup K of U_20 isomorphic to H. Let (G, . ) be a group and let H be a subgroup of H. Fix an element a of G and let K = { aha^-1 : h is subset of H} Prove that K is a subgroup of G. Prove that the map psi : H -> K, h -> aha^-1 is an isomorphism of groups. Let (G, . ) be a group and let a be an element of G. Dene the order of a. Give an example of an element of order 4 in GL(2,R). Find the order of 16 in Z_20

Here all fields are subfields of the complex numbers 0.1. Let F/K be a finite Galois extension and L/K a finite extension. Let FL be the smallest subfield of C containing both F and L. Show that FL/L is Galois and there is a natural injective homomorphism Gal(FL/L)Gal(F/K) Now we view the first group as a subgroup of the second. Identify Gal(FL/L) as Gal(F/K') for some subfield K CKCF (i.e. say exactly what K is, and of course prove your statement) Now assume L/K is also Galois. Show that Gal(FL/L) C Gal(F/K) is a normal subgroup, and identify the quotient group as a Galois group of some field extension. Show this quotient group is also a quotient of Gal(L/K), i.e. show there is a natural surjective homorphism Gal(L/K) ? Gal(F/K)/ Gal(FL/ L) 0.2. Let f KX be an rreducible polyn, whose Galois group is a non Abelian simple group (simple means it has no non-trivial normal subgroups). Use the results of problem (1) to prove that if L/K is a Galois extension with an Abelian Galois group, that f(x) E L[X] is irreducible, and has the same Galois group. Ie. Gal(f/L) = Gal(f/K). This is proven in Artin, Proposition 9.8 in the section on quintics. But his argument is overly complicated, if you repeat it, you will get some, but not full, credit Let g E K[X] be a quntic with Galois group either A5 or S5 and let L/K be an Abelian Galois extensions. Use (1) to show that the Galois group of g over L is either A5 or S5 (this includes the possibility that it started out S5 and then became A5). You may use the fact that A5 is simple, and that the only non-trivial proper normal subgroup of Ss is As. Prove that g(X) is irreducible over L. Now show if you have any tower of extensions with Li+1/L, an Abelian Galois extension, that g(X) E LIX] is irreducible

Let G be a group having a finite number of elements. Then For any a in G, there exist n in Z' such that a^n = e. for some n in Z^+, then G has n elements. Let b be the inverse of a in G. If there exists n in Z^+ such that a^n = e, then b ^n not equal e. If a is in G and a^n = e for some n in Z^+, then G is a cyclic group. Which of the following is true there exist a group in which the cancellation law holds. In every cyclic group every element is a generator. a cyclic group has a unique generator. Every group is a subgroup of itself. _______ Which of the following is a cyclic group The set of all integral multiples of 6 under addition. The set of 2 * 2 matrices having integer entries under addition. The set of symmetries of an equilateral triangle under function composition. The Klein 4-group. _________ Which of the following is false Every cyclic group is abelian. Not every abelian group is cyclic. Every group of order less than or equal to 4 is cyclic. If G is a cyclic group then G is isomorphic to Z under addition or to Z_s under addition modulo n where n is a positive integer. ______ Which of the following is false Every permutation is a one-to-one and onto function. The symmetric group S_3 is cyclic. The symmetric group S_10 has 10! Elements. Every subgroup of an abelian group is abelian.