Fall 2005 Math 411 Notes and Homework from Papantonopoulou's _Algebra: Pure & Applied_ (due each MONDAY - the *ed problems are for you to ponder - assignments are always subject to change as necessary): 9/12 Read Chapter 0, especially 0.3 on Z and mathematical induction. Note: Papa uses |X| for what I called #X, the cardinality of a set X. As you go through, try to do as many of the problems as you can, especially: 0.1 # 3,4,6,7,8,10,13,15,16,18 0.2 # 1,2,3,5,12,13 0.3 # 1,5,10*,11*,14,15,16,17,24,31,32; also work out (by induction) a formula for 1+3+5+...(2n-1) 0.4 # 1,8,11,15,19,25; also, compute (1+i)^8 0.5 # 4,7,10,15,17,18,20,21 9/19 1.1 # 3,5(cf.10),6,9,10(cf.5),12,13,18,21,23(for n=3,cf.12,13) 24, 26*,28*,29* 9/26 1.2 # 4,5,14,17,16,19,24,25,28,29,33,36 10/3 [Rosh Hashanah holidays 10/4-5, Ramadan begins 10/4] [No class Monday 10/3, makeup Friday 10/7] [HW due on Wednesday 10/5] 1.3 # 1,5,7,6,10,9,11*,13*,14*,15*,16,17,18*,19*,20,22*,24* 10/10 [Columbus Day 10/10, Yom Kippur holiday 10/13] [HW due on Wednesday 10/12] 1.4 # 4,5,7,9,10,13,14,15,19*,20*,21*,35(we talked about this Friday) 10/17 1.4 # 16,18,23*,24*,25*,26,29,30,32,34,36,38,39,40,41*,42*,43*,44 [Please finish the 1.4 hw, all to be submitted Monday 10/24] 10/24 2.1 # 2,8,9,10,11,12,18,20,21,22,23*,26*,27*,24*,25*,28*,29,30,31, 32*,33*,35*,37*,34*,39 10/31 2.2 # 2,3,4,6,14,16*,17*,18,24,28,33,34,36,40,42,48* 11/7 2.3 # 2,6,7,8,12,13,14*,15*,16*,29*,20,21,24,26*,27*,28* [The take-home midterm begins Wednesday 11/9 ... ====================================================================== 411 TAKE HOME MID-TERM Work alone - you may use your book or other published references, but you must cite these as any responsible scholar would - it is best if you work it out yourself, of course...! 1. Work out the groups (up to isomorphism) of low order, and exhibit the various groups of each order using a group-multiplication table: Prove that there is one group G of order 1, one with order 2, one with order 3, two with order 4, one with order 5, two with order 6 and one with order 7. How many G are there of order 8 (can you prove that these are all)? Indicate which of these G are abelian, and determine the centers Z(G) of those which are not. Now consider pairs of these groups G, G'(possibly G = G') and decide whether or not there is a nontrivial homomorphism G -> G' (give an example or prove there is none). 2. Work out and exhibit the subgroup graph of S_4, the group of all permutations of 4 elements (or equivalently, the rotation symmetries of a cube - the 4 "main diagonals" of the cube get permuted). Indicate which subgroups are normal in the bigger groups in which they are contained. 3. Let p be a prime, and let GL_n(Z/p) be the invertible n-by-n matrices with entries in Z/p; in other words, the determinant of any matrix in GL_n(Z/p) is invertible in Z/p, i.e. it is in U(p) = Z/p* = {1,2,...,p-1}. Let SL_n(Z/p) be those matrices with determinant 1. Show that SL_n(Z/p) is a normal subgroup of GL_n(Z/p) of index p-1, and that the quotient group GL_n(Z/p)/SL_n(Z/p) is isomorphic to U(p) = Z/p*. What is the order of the groups GL_2(Z/3) and SL_2(Z/3)? Can you figure out which groups these are isomorphic to? Can you find a general formula for the order in the n-by-n case with any p? 4. Show that H = = {1,-1,i,-i} is normal in G = Q_8 (the quaternion group). To what group is the quotient group G/H isomorphic? ====================================================================== 11/14 ... to be submitted Wednesday 11/16 - no class on Monday 11/14!] 11/21 [Thanksgiving holiday Thursday 11/24] 2.4 #3,10,16,19,22,27*,28*,29*,25*,26*,Cayley-(Di)Graphs problems* 2.5 #5,8,9,10,11,12,20*,21*,22* 11/28 3.1 #2,4,6,13,17,18* 3.2 #6,8,10,21* 3.3 #2,4,6,8,10* 12/5 3.4 #1,2,4,8,17,18 12/12 *****these are all optional - they aren't to be turned in***** 4.1 #1,5,6,7,10,11,12,13 4.2 #3,6,7,8,10,11 4.3 #1,3,8 4.4 Thm 4.4.4 (Class Equation) #1,8,10,11,12,13,14,15,16 4.5 Thm 4.5.4 (Conjugacy in S_n) #3,4,15 14 *****these are all optional - they aren't to be turned in***** Review [FINAL EXAM: 10:30 AM Saturday 12/17 LGRC A301 (the lowrise near science library)] ====================================================================== Introduction to Abstract Algebra Math 411 (Group Theory) Final Exam 10:30 AM 17 December 2005 LGRC A301 Rob Kusner 1. Warm-ups a. Suppose A and B are subgroups of C. If #A = 4 and #B = 9, then #(A \cap B) = _____ and #C is at least _____? b. If D is a group with #D = 29 then D \isom _____? c. If \phi: E \to F is a homomorphisms between groups of order #E = 28 and #F = 27, what is ker(\phi)? d. Determine all (non-isomorphic) abelian groups of order 900. 2. True/False a. G \isom Z for every nontrivial G < Z. b. If H < K are both abelian groups, then so is H/K. c. If L/M is an abelian group, then so are L and M. d. If N = N' \oplus N" and P = P' \oplus P", then N \times P = (N' \times P') \oplus (N" \times P"). 3. Symmetries The most symmetric letter "X" (without serifs) has a symmetry group of order 8, whereas adding a half-serif to one arm reduces the symmetries to the trivial group, and adding a full-serif to one arm gives a symmetry group of order 2. Explore the other decorations with full- or half-serifs and the corresponding symmetry groups. 4. Hard-core Cayley's Theorem implies that the cyclic group C_n can be realized as a subgroup of the symmetric group S_n. Show that we can do a bit better: for example, C_6 can be injected (1-to-1 homomorphism) into S_5 (why? hint: 2+3=5), and C_30 into S_10 (why? hint: 2+3+5=10). Find the smallest N(n) such that C_n injects into S_N(n). 5. Extra credit Consider the "rational numbers mod the integers" - this is the abelian group Q/Z under addition "mod 1"; for example 1/2 + 2/3 = 1/6 and 1/3 + 1/3 + 1/3 = 0. (i) Show that each g in Q/Z is of finite order. Unfortunately, Q/Z not finite, nor is it even finitely generated, so the fundamental theorem of finite (or finitely generated) abelian groups fails to apply. Here is a remedy: (ii) For each integer d > 1, let A(d) := {g = a/d^k}, that is, some power of d is in the denominator. For example, A(2) = {0, 1/2, 1/4, 3/4, 1/8, 3/8, 5/8, 7/8, ... , a/2^k, ...}. Show that: (a) A(d) is a subgroup of Q/Z. (b) A(c) \cap A(d) = {0} if gcd(c,d) = 1. (c) Q/Z = \oplus_{p prime} A(p). [Note A(p) is like an infinitely large version of Z/p^k: just as Z/p^k contains all of the *smaller* p-power cyclic groups as subgroups, our group A(p) contains *all* of the p-power cyclic groups!] ======================================================================