# Honors Introduction to Algebra -- Math 4340

## MWF 10:10-11:05 in Malott 224

Book: Dummit and Foote [DF] (parts I, IV, VI)

Office hours:

• mine, in my office Malott 515, directly after Monday classes
• Yun Liu's, Mondays at 4:15pm 4:30pm in Malott 218
The final exam is here, along with its LaTeX source if you need an example to get started with LaTeX. HW #1 (due Wednesday Jan 30)
• Find an isomorphism between the graphs and . (Number the vertices of each 1-10, such that i and j are connected in the first iff connected in the second.)
• How many isomorphisms are there?
• Find a graph with exactly 3 automorphisms (not more!).
• [DF] 1.1 #1,9,15
• HW #2 (due Wednesday Feb 5)
• [DF] 1.1 #25
• [DF] 1.2 #3,4
• [DF] 1.3 #2,14
• [DF] 1.7 #14,15,18
• HW #3 (due Wednesday Feb 12)
• [DF] 1.7 #23. ("Faithful" means that the hom from G to Sn is 1:1.)
• [DF] 2.1 #2,15
• [DF] 2.3 #3,14,20
• [DF] 2.4 #8
• [DF] 3.1 #7,24
• HW #4 (due Wednesday Feb 19). We'll prove that An is simple for n>4, with some warmup steps and then multiple stages of cornering our prey. Warning: this homework will be a little long, but not too hard.
• 1. Show that every cycle in Sn (any n) is a product of 2-cycles (i j), for various i,j and in some order.
• 2. Show that every element in Sn (any n) is a product of 2-cycles (i j).
• 3. Let N be a normal subgroup of Sn, containing some 2-cycle. Show that N = Sn.
• 4. Show that every element in An (any n) is a product in some order, of 3-cycles (i j k) and elements (a b)(c d).
• 5. Show that every element (a b)(c d) is the product of two 3-cycles.
• 6. If n>4, show that any two 3-cycles in An are conjugate.
• 7. Let N be a normal subgroup of An, containing some 3-cycle. Show that N = An.
• 8. Let pi be an even permutation with a 4-cycle (a b c d) (and possibly other cycles). Compute pi-1 o (b c d) o pi o (b d c).
• 9. Let pi be an even permutation with a cycle (a b c d e ...) of length >4 (and possibly other cycles). Compute pi-1 o (b c d) o pi o (b d c).
• 10. Let N be a normal subgroup of An, containing a permutation with a cycle of length 4 or more. Show that N = An.
• 11. Let pi be an even permutation with (at least) two 3-cycles (a b c)(d e f), and possibly other cycles. Compute pi-1 o (a b d) o pi o (a d b).
• 12. Let N be a normal subgroup of An, containing a permutation with (at least) two 3-cycles (a b c)(d e f), and possibly other cycles. Show that N = An.
• 13. If pi is an even permutation with no cycles of length > 3, what can its square look like?
• 14. Let pi be an even permutation with only 2-cycles, and at least three of them (a b)(c d)(e f). Compute pi-1 o (a e c) o pi o (a c e).
• 15. Let N be a normal subgroup of An, n > 5, and pi a non-identity element of N. Use the foregoing to help show these:
• If pi has a cycle of length 4 or more, show N = An. If not...
• If pi has two cycles of length 3, show N = An. If not...
• If pi has one cycle of length 3, the rest shorter, consider pi2 and show N = An. If not...
• If pi has only 2-cycles, but has three of them, show N = An. If not...
• If pi has only 2-cycles and fewer than three of them, show that N intersect A5 (considered as a subgroup of An) must be all of A5. Then, N must be all of An.
• HW #5 (due Wednesday Friday Feb 26)
• [DF] 4.1 #2,3
• [DF] 4.2 #8
• [DF] 4.3 #13,23,24,27
• HW #6 (due Wednesday Mar 12)
• Let G/H be a coset space (EDIT: with the usual G-action, on the left). Give a description of all the G-equivariant maps phi: G/H -> G/H. Do they form a group under composition?
• Let G be a group, all of whose elements are order 2. Show that G is isomorphic to a product of Z/2s. (EDIT: you may assume that G is finite. Or if you're feeling frisky, prove that this theorem is equivalent to the Axiom of Choice.)
• Let #G = p(p+1). Assume that there is >1 p-Sylow, and p>2.
• How many elements of G are not of order p? EDIT: nor of order 1
• Let S be a p-Sylow. Show that S acts transitively on that set of elements (not of order p).
• Conclude that they are all of order 2, and that p is Mersenne.
• Show that the 2-Sylow T is normal, and that G is a semidirect product of the Z/p acting on T.
• What are the p-Sylow subgroups of S9, for each p>2? Find an example for each p, and compute the normalizer of your example.
• HW #7 (due Wednesday Mar 19)
• Let G be the free group on the letters a..z. Impose relations that say homonyms in English are equal, for example to=too=two therefore w,o are trivial in this group. How small a group can you get?
• Let phi: H->N be a group homomorphism. Recall the homomorphism N -> Aut(N), n |-> (h |-> nhn-1). Let phi' : H -> Aut(N) be the composite. Use phi' to build a semidirect product G of H and N. Show that (even if phi' is nontrivial) G is isomorphic to the direct product of H and N.
• ...more to come...