Friday Nov 17

The number of p-Sylows is congruent to 1 mod p, and divides |G| / p^k.
Every p-subgroup is in a Sylow.
Sylow subgroups are all conjugate.
They're normal iff they're unique.
If |G|=2p, p an odd prime, then G is either Z_2p or D_p.

HW due Fri Nov 24Mon Nov 27

1. Let G be a p-group (i.e. |G|=p^k for p some prime). Use the action of G on itself by conjugation to show that G has nontrivial center.

2. Let G > H > N be a chain of groups, where N is normal in G. If H/N is normal in G/N, show that H is normal in G.

3. Let G be a p-group. Show that G has a normal subgroup of order p. Then show that G has subgroups of every order p^j dividing |G|. (Hint: a better statement is true -- it has a chain of normal subgroups.)

4. Find a 3-Sylow subgroup of each of S₆, S₉, S₁₅, and S₂₇. (If it helps, they have nice descriptions as semidirect products.)

5. Let p divide |G|, where |G| < p². Show that G has a normal p-Sylow. (Analytic number theory question: what fraction of numbers |G| have such p? Answer: log 2, or most of them.)

6. Let p^a || |G|, q^b || |G|, where p,q are prime. Assume that the number of q-Sylows is not a multiple of p. Show that G has a subgroup of order p^a q^b.

Wednesday Nov 15

Lagrange's thm. A₄ violates the converse statement.
p^k || m.
Def. Sylow subgroup, Syl_p(G).
Statement of Sylow's theorems.
Lemma. p doesn't divide (m choose p^k).
Lemma. If H,N finite subgroups of G, then |HN| = formula, and if H normalizes N, then HN is a subgroup.
Thm. Sylow subgroups exist. Pf. By acting on p^k-element subsets by left mult.
Thm. When S acts on Syl_p by conjugation, the only fixed point is itself.

HW due Fri Nov 17

1. Let G act diagonally on unordered pairs of group elements, g.{a,b} = {ga,gb}.
Describe the set of orbits, and the stabilizers, and (for G finite) the formula |pairs| = sum |orbits|.

[Some people asked whether a can equal b. Answer: the math doesn't particularly care. You may as well leave that possibility out, since it's kinda dumb. If you've already written this up with a=b allowed, that's fine too.]

2. Let S_n act on itself by conjugation. Explain how to compute the size of the conjugacy class of a permutation pi. What equation do you get for S₅?
(I.e. 5! = sum |conjugacy classes|, what are they?)

[H]
1.5 #18
1.6 #1,5,11

Monday Nov 6, Wednesday Nov 8, Monday Nov 13

k-transitivity and the Mathieu groups (for culture only).
Every transitive G-set is isomorphic to a coset space G/H.
Thm. Let G act on X finite. Then |X| = sum |G/Stab_i|, the stabilizers appearing.
Example: Let S_n act on the power set. Then 2ⁿ = sum (n choose k).
Cor. |G| = |X| |Stab|. Ex. |Rot(cube)| = #faces * (rotations of a face) = 6*4.

When is there a map from G/H -> G/K?
Classifying such. If H=K is finite, then they're automatically invertible.
Ex. Let Rot(cube) act on unordered pairs of vertices.
Then the orbits correspond to distance 1, distance 2, distance 3.
There is a Rot(cube)-equivariant map from {distance 1} -> {distance 3}, but none between any other pair of distinct orbits.

No class Friday (Veteran's Day, observed).

Let |G|=pⁿ act on X. Then |X| = |X^G| mod p.
Ex. G = Z_p acting on Fun(G,{1,...,n}) gives n^p = n mod p.
Center of a group.
Conjugacy classes in S_n.
In A_n they're more complicated: A₅ example.

HW due Fri Nov 10 Mon Nov 13

...since Nov 10 is a holiday.

1. Fix n>0 a natural number.
Call a function f:Z->Z is of period n if f(x+n) = f(x) + n for all x.
Let Jug_n denote the set of such f that are bijections.

A. Show that Jug_n is a group under function composition, i.e. the inverse of a function of period n is again of period n.
B. Show that Jug_n has an easy group homomorphism onto S_n, and that the kernel is isomorphic to Zⁿ.
C. Show that the map Jug_n ->> S_n has a one-sided inverse S_n -> Jug_n (a group homomorphism).
D. Infer that Jug_n is a semidirect product.

2. Let S_n act diagonally on {1,...,n}⁴, i.e. on 4-tuples.
How many orbits are there, as a function of n = 0,1,2,... ?

3. Say G,H are abelian, and G |X H is a semidirect product (i.e. H is the normal one). Assume that G is also normal. Show that G |X H is abelian.

4. Let Ad g:G -> G denote the function h |-> g h g^-1.
A. Show that Ad g is an automorphism of G.
B. Show that the map Ad: G -> Aut(G) is a group homomorphism.
C. What is its kernel?
D. Show that its image is a normal subgroup of Aut(G).

[H]
1.5 #6,11,16

Fri Nov 3

Transitive and simply transitive G-sets. G-equivariant maps and isomorphisms. G acting on a disjoint union of G-sets.
Theorem. Every G-set is G-equivariantly isomorphic to a disjoint union of transitive G-sets.
Free actions.
Theorem. If G acts freely on X finite, then |X| = |G| |X/G|.

Scary example: X = circle, G acts by antipode, X/G a circle again. Then the bijection used in the above theorem is not continuous. (Of course, continuity isn't very relevant when X is finite.)

Next time: we'll classify transitive G-sets and describes all morphisms between them.

Wednesday Oct 25 - Wednesday Nov 1

I had some delay creating this blog (Blogger kept being down or slow). In general I will try to keep these reasonably up-to-date.

Wed Oct 25

Definitions of group, homomorphism, kernel, subgroup, normal subgroup, product group, coset space, automorphism.
Definitions of Aut(G) for G a group, and semidirect product.

Mon Oct 30

Definition of group isomorphism. Recognition theorem for semidirect producst. Many examples thereof, including dihedral groups and the Euclidean group of motions. Short exact sequences of groups. Groups of order up to 10 (no proofs).

Wed Nov 1

Sym(X) = permutations of X. Two equivalent definitions of left G-set.
Right G-sets. Which, theorem, are equivalent to left G-sets (using group inverse).
G-orbits, X/G.
G acts on X, H acts on Y -> GxH acts on XxY. Diagonal action.

HW #1, due Nov 3

1. Let G be a group, N a subgroup. Show that the canonical projection G -> G/N can be made a group homomorphism for at most one group structure on G/N, and that there is such a group structure iff N is normal.

[Hungerford] problems:
Ch 1.2 #6,10 15bc
Ch 1.4 #6,9
Ch 1.5 #1,7,12,14

Math 200: Graduate Algebra

I'll be putting homeworks and day-to-day synopses here.