Compute the size of the automorphism group of the graph which is the 1-skeleton of the 3-dimensional cube.
Describe the size of each conjugacy class in A_4.
Consider a graph P_6 with vertices all parenthetizations of ABCDE and edges all associativities. It has 14 vertices. One of them is (AB)((CD)E) for example. Draw the graph P_6 as the 1-skeleton of a convex 3-dimensional polytope.
Problem 1: Recall that SL(n,q) is the group of n by n matrices with entries in a finite field of q elements with determinant 1. PSL(n,q) is the quotient of SL(n,q) with the diagonal matrices with equal entries. Show that PSL(2,4) and PSL(2,5) are simple groups of 60 elements. Conclude, using a theorem of the class, that they are isomorphic to A_5.
Problem 2: Show that PSL(2,2) is isomorphic to S_3 and PSL(2,3) is isomorphic to A_4 are not simple groups.
Problem 3: Show that if G has exactly 1+kp p-Sylow subgroups then there exist a subgroup H of S_{1+kp} having exactly 1+kp p-Sylow subgroups.
Problem 4: Given p=prime, construct an example of a finite group with p+1 p-Sylow subgroups.
Problem 1: Compute a basis for the kernel Ker(A) and the type of the finitely generated abelian group coker(A) where A=( 1 2 3 4 \\ 5 6 7 8 \\ 9 10 11 12) is a 3x4 matrix with first column (1 2 3 4).
Problem 1. Let Z denote the center of the group-ring Q[G] of a finite group G.
(a) Show that the sum C of the elements of a fixed conjuugacy class is in Z.
(b) Show that Z is a 3-dimensional vector space over Q when G=Sym_3.
(b) Compute explicitly generators x_i i=1,2,3 of Z in (b) and express their product x_i.x_j explicitly as a linear combination of x_k.
2. Let C[[x]] denote the set of formal power series in x with complex coefficients. Prove that (x) is a maximal ideal.
What are all the maximal ideals of C[[x]]?
What are the maximal ideals of C[[x,y]]?
3. Let R be a commutative ring and I an ideal of R. We define the radical rad(I) to be the set of all x in R so that x^m is in I for some m>0. Prove that rad(I) is an ideal. Compute the rad(108 Z) where 108 Z is an ideal of the integers Z.
4. Find an example of an irreducible polynomial in Z[x] which becomes reducible in Z/3Z[x].
Homework 9 due Tuesday 11/10.
Problems 8,9,10,28,29 from chapter 3. Hint for problem 10: given m in M look at ann(m)={a in A so that am=0}. Show that ann(m) is an ideal and m is nonzero iff ann(m) is not A.
Problem: An example of a filtered algebra A is the following. Consider the noncommutative algebra over Q generated by x,y,h modulo the 2-sided ideal I generated by xy-yx=h, hx-xh=2x,hy-yh=-2y. Let A_n denote the Q-span of all words in x,y,h of length at most n. Show that this defines a filtration on A. Show that gr(A) is isomorphic to Q[x,y,h]. The algebra A is the universal enveloping algebra of the Lie algebra sl_2.
Homework 10 due Tuesday 11/17.
Problems 1,5,18,19,20 from chapter 4, p213.
Compute the resultant of ax^2+bx+c and dx^3+ex^2+fx+g. You may wish to confirm your answer using Mathematica.
Homework 11 due Tuesday 11/24.
Problems 1,2,3,4,5,6,7 fro chapter V p.253.