Math 521, Lecture 2
Hints for Assignment # 3
Problem 1 Let X be a metric space, and let E X be a subset. The interior of the set E is the
set E o = x E there exists r > 0 so that B(x, r) E . The closure of E is the set E = E E
where E is the set of limit
2. Groups I
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Groups
Subgroups, Lagranges theorem
Homomorphisms, kernels, normal subgroups
Cyclic groups
Quotient groups
Groups acting on sets
The Sylow theorem
Trying to classify nite groups, part I
Worked examples
1. Gr
I. Algebra
1. Vector Spaces
1A. Linearity.
Denition 1.1. A real vector space (or just a vector space for short) consists of a
set V , a function V V V called addition, and a function R V V called scalar
multiplication.
For v, w V , we write v + w for the
HOMOMORPHISMS
KEITH CONRAD
1. Introduction
In group theory, the most important functions between two groups are those that preserve the
group operations, and they are called homomorphisms. A function f : G H between two groups
is a homomorphism when
f (xy
WON Series in Discrete Mathematics and Modern Algebra Volume 2
NUMBER THEORY
Amin Witno
Preface
Written at Philadelphia University, Jordan for Math 313, these notes1 were used rst
time in the Fall 2005 semester. They have since been revised2 and shall be
MAT 364 Topology
Problem Set 6
due Wednesday, October 13
Problem 1. Determine whether the following sets are compact or not.
Prove your answers.
(a) X = cfw_(x, y) R2 |x 0, y 0, the graph of the function y = sin x.
Solution. Not compact since its not boun
MATH 115, SUMMER 2012
LECTURE 13
JAMES MCIVOR
Today we focus on the multiplicative aspects of the integers mod p.
1. Primitive roots in Z/m
For this section, we work in the ring Z/m, so = in this ring means congruent
mod m in integers. We are interested i
Stochastic Processes
Section 6.1
Section 6.2
Homework
Chapter 6: The Bernoulli and Poisson Processes
MATH/STAT 395: Probability II
Roddy Theobald
Summer 2014
1 / 24
Stochastic Processes
Section 6.1
Section 6.2
Homework
Stochastic Processes
Each of these s
Solution to last chapter 5 activity problem
As we showed in step 3, for a given sample size n, the null hypothesis is rejected if
1.645. We want the probability of this event (the power) to be .95:
Mn 280
14.2/ n
P
Mn 280
1.645
14.2/ n
= .95
We also kno
SUBGROUP SERIES II
KEITH CONRAD
1. Introduction
In part I, we met nilpotent and solvable groups, dened in terms of normal series. Recalling the denitions, a group G is called nilpotent if it admits a normal series
(1.1)
cfw_e = G0
G1
G2
Gr = G
in which Gi