Problem 1E: Which of the following binary operations are closed? a. subtraction of positive integers b. division... Problem 2E: Which of the following binary operations are associative? a. subtraction of integers b. division of... Problem 3E: Which of the following binary operations are commutative? a. substraction of integers b. division of... Problem 4E: Which of the following sets are closed under the given operation? a. {0, 4, 8, 12} addition mod 16... Problem 5E: In each case, find the inverse of the element under the given operation. a. 13 in Z20 b. 13 in U(14)... Problem 6E: In each case, perform the indicated operation. a. In C*, (7+5i)(3+2i) b. In GL(2,Z13) , det [7415]... Problem 7E Problem 8E: List the elements of U(20). Problem 9E: Show that {1, 2, 3} under multiplication modulo 4 is not a group but that {1, 2, 3, 4} under... Problem 10E: Show that the group GL(2,R) of Example 9 is non-Abelian by exhibitinga pair of matrices A and B in... Problem 11E: Let a belong to a group and a12=e . Express the inverse of each of the elements a, a6,a8,anda11 in... Problem 12E: In U(9)find the inverse of 2, 7, and 8. Problem 13E: Translate each of the following multiplicative expressions into itsadditive counterpart. Assume that... Problem 14E: For group elements a, b, and c, express (ab)3and(ab2c)2 withoutparentheses. Problem 15E: Suppose that a and b belong to a group and a5=eandb7=e .Write a2b4and(a2b4)2 without using negative... Problem 16E: Show that the set {5, 15, 25, 35} is a group under multiplication modulo 40. What is the identity... Problem 17E: Let G be a group and let H=x1xG . Show that G=H as sets. Problem 18E: List the members of K=x2xD4andL=xD4x2=e . Problem 19E: Prove that the set of all 22 matrices with entries from R and determinant +1 is a group under matrix... Problem 20E: For any integer n2 , show that there are at least two elements inU(n) that satisfy x2=1 . Problem 21E: An abstract algebra teacher intended to give a typist a list of nine integers that form a group... Problem 22E: Let G be a group with the property that for any x, y, z in the group, xy=zx implies y=z . Prove that... Problem 23E: (Law of Exponents for Abelian Groups) Let a and b be elements of an Abelian group and let n be any... Problem 24E: (SocksShoes Property) Draw an analogy between the statement (ab)1=b1a1 and the act of putting on and... Problem 25E: Prove that a group G is Abelian if and only if (ab)1=a1b1 for all a and b in G. Problem 26E: Prove that in a group, (a1)1=a for all a. Problem 27E: For any elements a and b from a group and any integer n, prove that (a1ba)n=a1bna . Problem 28E: If a1,a2,...,an belong to a group, what is the inverse of a1,a2,...,an ? Problem 29E: The integers 5 and 15 are among a collection of 12 integers that form a group under multiplication... Problem 30E Problem 31E Problem 32E: Construct a Cayley table for U(12). Problem 33E: Suppose the table below is a group table. Fill in the blank entries. Problem 34E: Prove that in a group, (ab)2=a2b2 if and only if ab=ba . Prove that in a group, (ab)2=b2a2 if and... Problem 35E: Let a, b, and c be elements of a group. Solve the equation axb=c for x. Solve a1xa=c for x. Problem 36E: Let a and b belong to a group G. Find an x in G such that xabx1=ba. Problem 37E: Let G be a finite group. Show that the number of elements x of Gsuch that x3=e is odd. Show that the... Problem 38E: Give an example of a group with elements a, b, c, d, and x such that axb=cxd but abcd . (Hence... Problem 39E: Suppose that G is a group with the property that for every choice of elements in G, axb=cxd implies... Problem 40E: Find an element X in D4 such that R90VXH=D . Problem 41E: Suppose F1andF2 are distinct reflections in a dihedral group Dn .Prove that F1F2R0 . Problem 42E: Suppose F1andF2 are distinct reflections in a dihedral group Dn such that F1F2=F2F1. Prove that... Problem 43E: Let R be any fixed rotation and F any fixed reflection in a dihedral group. Prove that RkFRk=F . Problem 44E: Let R be any fixed rotation and F any fixed reflection in a dihedral group. Prove that FRkF=Rk . Why... Problem 45E: In the dihedral group Dn , let R=R360/n and let F be any reflection.Write each of the following... Problem 46E: Prove that the set of all 33 matrices with real entries of the form [ 1 a b 0 1 c 0 0 1] is a group.... Problem 47E: Prove that if G is a group with the property that the square of every element is the identity, then... Problem 48E: In a finite group, show that the number of nonidentity elements that satisfy the equation x5=e is a... Problem 49E: List the six elements of GL(2,Z2) . Show that this group is non-Abelian by finding two elements that... Problem 50E: Prove the assertion made in Example 19 that the set 1,2,...,n1 is a group under multiplication... Problem 51E: Suppose that in the definition of a group G, the condition that there exists an element e with the... Problem 52E: Suppose that in the definition of a group G, the condition that for each element a in G there exists... format_list_bulleted