Abstract Algebra I Notes
UIUC MATH 500, F’08
Jingjin Yu
1
Groups
08/25/08  08/27/08
Definition 1.1
Semigroups, monoids, groups, rings and commutative rings. For a map
G
G
G
, consider
the following properties:
1) Closure,
2) Association,
3) Identity,
4) Inverse,
5) Commutative.
A set
G
with a composition law
G
G
G
is called :
a
semigroup
if it satisfies 12,
a
monoid
if it satisfies 13,
a
group
if it satisfies 14,
a
abelian group
if it satisfies 15.
Definition 1.2
A
subgroup
H
of group
G
is a subset of
G
that is also a group with the same map of
G
G
G
.
Example 1.3
0: semigroup;
0: monoid;
: group.
Definition 1.4
A group
homomorphism
is a function
f
:
G
H
s.t.
f xy
f x f y
. If
f
is injective,
surjective, and bijective, then we call them
monomorphism
,
epimorphism
,
isomorphism
.
Example 1.5
,
,
is an injective homomorphism; but there does not exist a nonzero homomor
phism
,
,
. Suppose not then
f
1
0 and
nf
1
n
f n
1
n
f
1
f
1
n
0. But no
matter what
f
1
is, it cannot be infinity and there are some
n
such that 0
f
1
n
f
1
n
1. This is
not possible since
f
1
n
.
Definition 1.6
A permutation of a set
X
is a bijection
X
X
:
1
. . . . . .
n
a
1
. . . . . .
a
n
1
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Definition 1.7
The
symmetric group
on a set
X
is the collection of all permutations on
X
with notation
Symm
X
. If
X
1
, . . . , n
, we write
S
n
for Symm
X
.
Remark.
We have injection
φ
:
S
n
S
n
1
by extending any permutation
f
with
f n
1
n
1.
We may define
S
n
1
S
n
.
If we let
X
1
,
2
, . . .
, then
S
Symm
X
since the permutation
σ
2
,
1
,
4
,
3
,
6
,
5
. . .
Symm
X
, but
σ
S
.
Theorem 1.8
Let
f
:
G
H
be a homomorphism.
(1)
f
is a monomorphism if and only if ker
f
e
.
(2)
f
is an isomorphism if and only if there eixsts a homomorphism
f
1
s.t.
f
f
1
1
H
, f
1
f
1
G
.
Proof.
(1) (
) Clear. (
) Suppose
f x
f y
, then
f xy
1
f
1
1
H
, by assumption
xy
1
1
x
y
.
(2) (
)
f
is invertible, we just need to verify that it is a homomorphism:
f
1
ab
f
1
a f
1
b
. (
)
Clear.
08/29/08
Definition 1.9
let
a
1
, . . . , a
r
be distinct elements of
X
1
,
2
, . . . , n
, and let
Y
X
a
1
, . . . , a
r
. If
f
S
n
fixes every element in
Y
and
f a
i
a
i
1
for
i
1
, r
1 and
f a
r
a
1
, then
f
is called an
r
cycle
and we
write
f
a
1
, . . . , a
r
. A 2cycle is called a
transposition
.
Definition 1.10
Two permutations
α, β
are disjoint if
(1)
α k
k
β k
k
, and
(2)
β k
k
α k
k
.
Proposition 1.11
Every permutation
α
S
n
is a composite (product) of disjoint cycles.
Proof.
By induction on the number of elements that moved by
α
.
(case
m
0)
α
1
is the identity.
(case
m
0) Choose
a
1
with
α a
1
a
1
and let
a
2
α a
1
, . . . a
i
1
α a
i
until we have
a
l
a
1
, . . . , a
l
1
.
We claim that
a
l
a
1
.
Suppose not and
a
l
a
i
,
1
i
l
, then
a
l
α a
l
1
and
a
i
α a
i
1
, which imply that
a
l
1
a
i
1
, contradicting that
a
l
is the first repetition.
We may then get a cycle that is disjoint from the rest of the permutation and apply induction hypothesis.
Definition 1.12
G
is a group and
a, g
G
, then
gag
1
is a conjugate of
a
.
Example 1.13
G
S
n
, let
α
a
1
, . . . , a
n
S
n
, then for
τ
S
n
, τατ
1
τ a
1
, . . . , τ a
n
.
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