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 misleading specics. Specically,
introduce the nonabelian group
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 normal subgroup of G.
Proof. The group G acts on the coset s
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 element acts trivially,
1G x = x
for all x S.
Some examples
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 1 < d1 | d2 | | dt ,
a nonnegative integer r
such that
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 literally no dierence between them. The idea is to dene the
p
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(f ),
respectively a normal subgroup of G and a subgrou
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 H equivalent if they produce the same left coset of H ,
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 can be written (nonuniquely) as a product of
transposi
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 commutator of any two elements a and b of G is dened as
[a,
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, as a description of solving a problem, so that
decomp
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 cyclic if it is generated by one element, i.e., if it tak
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 the set
of integers is notated
Z = cfw_0, 1, 2, ,
with
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 is, the map f must satisfy the condition
f (g G g ) = f
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 two-sided: Given a,
let b a right-inverse of a and then
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 as
n = pe1 per ,
r
1
r 0, p1 , , pr P , e1 , , er Z+ ,