18.317 Combinatorics, Probability, and Computations on Groups
22 October 2001
Lecture 18
Lecturer: Igor Pak
Scribe: Christopher Malon
Testing Solvability and Nilpotence
How to Reduce Generating Sets
Let G = g1 , . . . , gk , and consider a random subprodu
18.317 Combinatorics, Probability, and Computations on Groups
17 October 2001
Lecture 14
Lecturer: Igor Pak
Scribe: C. Goddard
Hall Bases Continued
Last lecture we nished with the theorem:
Theorem 1. Given a - complete word in B = (B1 , B2 , . . .), a Hal
18.317 Combinatorics, Probability, and Computations on Groups
16 November 2001
Lecture 26
Lecturer: Igor Pak
Scribe: Igor Pavlovsky
Babais Algorithm continued: escape time
Last time, we proved
Theorem 1 Let C be a subset of the group G, S = S 1 a symmetri
18.317 Combinatorics, Probability, and Computations on Groups
28 November 2001
Lecture 30
Lecturer: Igor Pak
Scribe: Michael Korn
Bias in the Product Replacement Algorithm
Here is the algorithm:
The input to the algorithm is a k-tuple g = (g1 , g2 , . . .
18.317 Combinatorics, Probability, and Computations on Groups
30 October 2001
Lecture 31
Lecturer: Igor Pak
Scribe: D. Jacob Wildstrom
Bias (continued)
Theorem 1 (P. Hall). For a simple group H and G = H m , it follows that g1 , . . . , gk = G if and only
18.317 Combinatorics, Probability, and Computations on Groups
15 October 2001
Lecture 15
Lecturer: Igor Pak
Scribe: Fumei Lam
Hall Bases
Denition 1 Let H be an abelian p-group. A set B = cfw_b1 .b2 , . . . br H is a Hall basis if
h H, 1 , 2 , . . . r cfw
18.317 Combinatorics, Probability, and Computations on Groups
10 October 2001
Lecture 13
Lecturer: Igor Pak
Scribe: Bo-Yin Yang
More on Strong Uniform Stopping Times
Theorem 1 We have a tight bound on group random walks with a strong uniform stopping time
18.317 Combinatorics, Probability, and Computations on Groups
October 29, 2001
Lecture 0
Lecturer: Igor Pak
Scribe: M. Alekhnovich
Mixing time & long paths in graphs
Let be a Cayley graph of group G: = Caley(G, S), |S| = D, | = n. Recall the following not
18.317 Combinatorics, Probability, and Computations on Groups
26 November 2001
Lecture 29
Lecturer: Igor Pak
Scribe: Etienne Rassart
Two theorems on the product replacement graph
Let k (G) be the graph with vertex set cfw_(g1 , . . . , gk ) Gk : g1 , . .
18.317 Combinatorics, Probability, and Computations on Groups
5 November 2001
Lecture 22
Lecturer: Igor Pak
Scribe: Nate Ackerman
Theorem 1 (This is a special case of the theorem from last class.)
Suppose = (G, S), = set of paths in , and = cfw_x : path f
18.317 Combinatorics, Probability, and Computations on Groups
21 November 2001
Lecture 28
Lecturer: Igor Pak
Scribe: Fumei Lam
Product Replacement Graphs
Denition 1 Let G be a nite group and let k d(G), where d(G) is the minimum number of generators of G.
18.317 Combinatorics, Probability, and Computations on Groups
12 October 2001
Lecture 14
Lecturer: Igor Pak
Scribe: D. Jacob Wildstrom
Random Walks on Nilpotent Groups
Example 1. Let
1
0 1
G = . . .
.
. .
. .
0 0
. Fp ,
.
.
1
that is to say, the gr
18.317 Combinatorics, Probability, and Computations on Groups
November 2, 2001
Lecture 21
Lecturer: Igor Pak
1
Scribe: B. Virag
Dirichlet forms and mixing time
Let G be a nite group, and let V be the vector space of real-valued functions from G. There is