MATHEMATICS 152, FALL 2003METHODS OF DISCRETE MATHEMATICSHomework Assignment # 4Due: October 14, 2003ReadingRead Biggs, Chapter 22.Required Problems1. Show that H = {I, (12)} is not a normal subgroup of S4by computingits conjugate subgroups.2. Show that J = {I, (123), (132)} is a normal subgroup of S3by computingits conjugate subgroups. What is the quotient group S3/J?3. Consider the group G = (Z×13, ⊗) and the subgroup H generated by [4].Determine the quotient group G/H by writing down its elements (thecosets) and writing a group table for them.4. Consider the ring R = M2(Z2), that is, the 2 × 2 matrices with entriesfrom the field Z2.(a) How many elements does R have?(b) Find all elements that have multiplicative inverses, and list themwith their inverses.5. Consider the ring of Gaussian integers, R = Z[i] = {a + bi|a, b ∈ Z},where i2= −1. With the usual addition and multiplication from thecomplex numbers, it may be shown that R is a ring. Which elementshave multiplicative inverses, and what are they?6. Problem #22.1.3 in Biggs: Show that if x and y are me mbers of a ring Rthen (−x)(y) = −(xy) and (−x)(−y) = xy. At each stage of the proof,explain which property of R you are using.1Exploratory Problem7. In this exercise, we consider the group A4and its normal subgroup V4, theKlein Four group, and show that the quotient group A4/V4is isomorphicto C3.Exhibit V4as a (normal) subgroup of the rotation group of the tetra-hedron. List as permutations the members of the three cosets. Identifyeach coset with an element of C3. For each pair of cosets (recall that C3is abelian!), perform the multiplication by members of the cosets chosenat random, and thereby reconstruct the multiplication table for