# Honors Introduction to Algebra -- Math 4340

## Prof. Allen Knutson

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

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

This page is out of date; see the Canvas page.

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)
HW #2 (due Wednesday Feb 5)
HW #3 (due Wednesday Feb 12)
HW #4 (due Wednesday Feb 19). We'll prove that A_{n} 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.
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 S_{9}, 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...