INTRODUCTORY LECTURE ON GROUPS
1. Motivate Isomorphism Immediately
The same group can appear in various manifestations, so it behooves us to see
the underlying common structure rather than the mislead
ANALYSIS OF SMALL GROUPS
1. Big Enough Subgroups are Normal
Proposition 1.1. Let G be a nite group, and let q be the smallest prime divisor
of |G|. Let N G be a subgroup of index q . Then N is a norma
THE SYLOW THEOREMS
1. Group Actions
An action of a group G on a set S is a map
G S S,
(g, x) gx
such that
The action is associative,
(g g )x = g (x)
g
for all g, g G and x S.
The group identity elem
FINITELY-GENERATED ABELIAN GROUPS
Structure Theorem for Finitely-Generated Abelian Groups. Let G be a
nitely-generated abelian group. Then there exist
a nonnegative integer t and (if t > 0) integers
GROUP PRODUCTS
Many beginning group theory texts distinguish between the external direct product and the internal direct product of groups. This writeup explains a viewpoint
from which there is litera
THE THREE GROUP ISOMORPHISM THEOREMS
1. The First Isomorphism Theorem
Theorem 1.1 (An image is a natural quotient). Let
f : G G
be a group homomorphism. Let its kernel and image be
K = ker(f ),
H = im
COSETS IN LAGRANGES THEOREM AND IN GROUP
ACTIONS
1. Lagranges Theorem
Let G be a group and H a subgroup, not necessarily normal.
Denition 1.1 (Left H -equivalence). Two group elements g, g G are left
THE ALTERNATING GROUP IS SIMPLE
1. Some Properties of the Symmetric Group
Let n Z>0 be a a positive integer, and let Sn denote the group of permutations
of n letters.
The fact that every element of Sn
ABELIANIZING A GROUP
Let G be a group. Its centralizer, Z (G), is an abelian normal subgroup. We also
would like an abelian quotient of G that retains as much information about G as
possible.
The comm
KERNELS AND QUOTIENTS
A hopelessly broad problem is:
Classify all groups.
A variant of the problem is:
Given a group, try to study it by breaking it into smaller pieces.
A group can arise, for example
CYCLIC GROUPS
1. Definition
Recall that if G is a group and S is a subset of G then the notation
S
signies the subgroup of G generated by S , the smallest subgroup of G that contains S .
A group is cy
THE INTEGERS
1. Divisibility and Factorization
Without discussing foundational issues or even giving a precise denition, we
take basic operational experience with the integers for granted. Specically
GROUP ACTIONS
1. Review of Homomorphisms
Recall that if (G, G ) and (G, G ) are groups then a set-map
e
f : G G
is a homomorphism if the following diagram commutes:
GG
(f,f )
/ GG
G
e
G
G
/ G.
f
That
BASIC FACTS ABOUT GROUPS
(To be lled in later.)
Uniqueness of identity: e = e e = e.
Uniqueness of inverse: b = be = b(ac) = (ba)c = ec = c.
Granting a right-identity and right-inverses, they are t
EUCLIDEAN RINGS
1. Introduction
The topic of this lecture is eventually a general analogue of the unique factorization theorem for the integers:
Theorem 1.1. Let n be a nonzero integer. Then n factors