18.317 Combinatorics, Probability, and Computations on Groups
1 October 2001
Lecture 10
Lecturer: Igor Pak
Scribe: Igor Pavlovsky
Coupon Collectors Problem
Recall that the problem concerns a prudent shopper who tries, in several attempts, to collect a com
18.317 Combinatorics, Probability, and Computations on Groups
5 October 2001
Lecture 12
Lecturer: Igor Pak
Scribe: Michael Korn
More About Mixing Times
Denition 1 For a probability distribution Q, we dene the total variation distance, denoted Q U , by
Q U
18.317 Combinatorics, Probability, and Computations on Groups
28 September 2001
Lecture 9
Lecturer: Igor Pak
Scribe: Christopher Malon
Generalized Random Subproducts
Today, we start to upgrade the ErdsRnyi machine to show that most generating sets of size
18.317 Combinatorics, Probability, and Computations on Groups
14 September 2001
Lecture 4
Lecturer: Igor Pak
Scribe: Ben Recht
Even if a paper is famous and written by very famous individuals, that does not necessarily mean that it
is correct. In this lec
18.317 Combinatorics, Probability, and Computations on Groups
3 October 2001
Lecture 11
Lecturer: Igor Pak
Scribe: N. Ackerman
1. Random Walks on Groups
Let G be a nite Group and let S be a set of generators of G. We say that S is
symmetric if S = S 1 . I
18.317 Combinatorics, Probability, and Computations on Groups
12 September 2001
Lecture 3
Lecturer: Igor Pak
Scribe: T. Chiang
Probabilistic Generation
In this lecture, we will prove the following Theorem with contemporary mathematics:
Theorem 1 (Dixon)
P
18.317 Combinatorics, Probability, and Computations on Groups
19 September 2001
Lecture 5
Lecturer: Igor Pak
Scribe: Dennis Clark
Proof of a Lemma
Our plan here is to prove the following lemma, needed in the xed proof of Erds-Turans theorem:
o
Lemma 1 Sup
18.317 Combinatorics, Probability, and Computations on Groups
26 September 2001
Lecture 8
Lecturer: Igor Pak
Scribe: Bo-Yin Yang
The Plot
Our general plan for these few lectures is to prove the following will usually (with probability arbitrarily
close to
18.317 Combinatorics, Probability, and Computations on Groups
21 September 2001
Lecture 6
Lecturer: Igor Pak
Scribe: C. Goddard
Probabilistic Generation
In this lecture, we will complete the classical proof for Dixons theorem on the probabilistic generati
18.317 Combinatorics, Probability, and Computations on Groups
10 September 2001
Lecture 2
Lecturer: Igor Pak
Scribe: Jason Burns
The probability of generating a group, part 2
Some notation from last time:
Let G be a group. We will write k (G) for the prob
18.317 Combinatorics, Probability, and Computations on Groups
7 September 2001
Lecture 1
Lecturer: Igor Pak
Scribe: R. Radoii
cc
Probability of Generating a Group
Let G be a nite group and let |G| denote the order of G. Let d(G) denote the minimum number
18.317 Combinatorics, Probability, and Computations on Groups
24 September 2001
Lecture 7
Lecturer: Igor Pak
Scribe: Etienne Rassart
The Erds-Rnyi Machine
o
e
Let g1 , g2 , . . . , gk G and consider h = g1 1 gk k , where the i cfw_0, 1 are i.i.d. random v