00305 - Group Theory [Milne].pdf

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GROUP THEORY
J.S. MILNE
August 21, 1996; v2.01
Abstract. Thes are the notes for the first part of Math 594, University of Michigan, Winter
1994, exactly as they were handed out during the course except for some minor corrections.
Please send comments and corrections to me at jmilne@umich.edu using “Math594” as
the subject.
Contents
1. Basic Definitions
1
1.1. Definitions
1
1.2. Subgroups
3
1.3. Groups of order < 16
4
1.4. Multiplication tables
5
1.5. Homomorphisms
5
1.6. Cosets
6
1.7. Normal subgroups
7
1.8. Quotients
8
2. Free Groups and Presentations
10
2.1. Free semigroups
10
2.2. Free groups
10
2.3. Generators and relations
13
2.4. Finitely presented groups
14
The word problem
The Burnside problem
Todd-Coxeter algorithm
Maple
3. Isomorphism Theorems; Extensions.
16
3.1. Theorems concerning homomorphisms
16
Factorization of homomorphisms
The isomorphism theorem
The correspondence theorem
Copyright 1996 J.S. Milne. You may make one copy of these notes for your own personal use.
i
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ii
J.S. MILNE
3.2. Products
17
3.3. Automorphisms of groups
18
3.4. Semidirect products
21
3.5. Extensions of groups
23
3.6. The Holder program.
24
4. Groups Acting on Sets
25
4.1. General definitions and results
25
Orbits
Stabilizers
Transitive actions
The class equation
p -groups
Action on the left cosets
4.2. Permutation groups
31
4.3. The Todd-Coxeter algorithm.
35
4.4. Primitive actions.
37
5. The Sylow Theorems; Applications
39
5.1. The Sylow theorems
39
5.2. Classification
42
6. Normal Series; Solvable and Nilpotent Groups
46
6.1. Normal Series.
46
6.2. Solvable groups
48
6.3. Nilpotent groups
51
6.4. Groups with operators
53
6.5. Krull-Schmidt theorem
55
References:
Dummit and Foote, Abstract Algebra.
Rotman, An Introduction to the Theory of Groups
GROUP THEORY
1
1. Basic Definitions
1.1. Definitions.
Definition 1.1. A is a nonempty set G together with a law of composition ( a,b )
→
a
∗
b :
G × G → G satisfying the following axioms:
(a) (associative law) for all a,b,c
∈
G ,
( a
∗
b )
∗
c = a
∗
( b
∗
c );
(b) (existence of an identity element) there exists an element e
∈
G such that a
∗
e = a =
G ;
(c) (existence of inverses) for each a
∗
a for all a
∈
∈
G , there exists an a ∈
G such that
a
∗
a = e = a ∗
a.
If (a) and (b) hold, but not necessarily (c), then G is called a semigroup . (Some authors
don’t require a semigroup to contain an identity element.)
We usually write a ∗ b and e as ab and 1, or as a + b and 0.
Two groups G and G are isomorphic if there exists a one-to-one correspondence a
↔
a ,
G ↔ G , such that ( ab ) = a b for all a,b ∈ G .
Remark 1.2. In the following, a,b,... are elements of a group G .
(a) If aa = a ,then a = e (multiply by a ). Thus e is the unique element of G with the
property that ee = e .
(b) If ba = e and ac = e ,then
b = be = b ( ac )=( ba ) c = ec = c.
Hence the element a in (1.1c) is uniquely determined by a .Wecallitthe inverse of a ,and
denote it a − 1 (or the negative of a , and denote it −a ).
(c) Note that (1.1a) allows us to write a 1 a 2 a 3 without bothering to insert parentheses.
The same is true for any finite sequence of elements of G . For definiteness, define
a 1 a 2
···
a n =(
···
(( a 1 a 2 ) a 3 ) a 4
···
) .
are inserted. (See Dummit p20.) Thus, for any finite ordered set S of elements in G , a∈S a
is defined. For the empty set S , we set it equal to 1.
(d) The inverse of a 1 a 2 ···a n is a − 1
1 .
(e) Axiom (1.1c) implies that cancellation holds in groups:
n a − 1
n− 1
···a − 1
b = c
(multiply on left or right by a − 1 ). Conversely, if G is finite , then the cancellation laws imply
Axiom (c): the map x
ab = ac =
⇒
b = c, ba = ca =
⇒
G is injective, and hence (by counting) bijective; in
particular, 1 is in the image, and so a has a right inverse; similarly, it has a left inverse, and
we noted in (b) above that the two inverses must then be equal.
→
ax : G
→
The order of a group is the number of elements in the group. A finite group whose order
is a power of a prime p is called a p-group. 1
1 Throughout the course, p will always be a prime number.
e
Then an induction argument shows that the value is the same, no matter how the parentheses
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J.S. MILNE
Define
aa
···
a n> 0 n copies)
a n =
1
n =0
a − 1 a − 1
···
a − 1
n< 0 n copies)
The usual rules hold:
(1.1)
It follows from (1.1) that the set
a m a n = a m + n , ( a m ) n = a mn .
}
is an ideal in Z . Therefore, this set equals ( m )forsome m ≥ 0. If m =0,then a is said to
have infinite order ,and a n
{n ∈ Z | a n =1
= 1 unless n =0. Otherwise, a is said to have finite order m,
and m is the smallest positive integer such that a m =1. Inthiscase, a n =1 ⇐⇒
m|n ;
moreover a − 1 = a m− 1 .
Example 1.3. (a) For each m =1 , 2 , 3 , 4 ,... ,∞ there is a cyclic group of order m , C m .
When m<∞ , then there is an element a ∈ G such that
G = { 1 ,a,...,a m− 1
}
,
and G can be thought of as the group of rotations of a regular polygon with n -sides. If
m =
∞
, then there is an element a
∈
G such that
G =
{
a m
|
m
∈ Z }
.
,and a is called a generator of C m .
(b) Probably the most important groups are matrix groups. For example, let R be a
commutative ring 2 .If X is an n
≈ Z
/m
Z
n matrix with coeCcients in R whose determinant is
a unit in R , then the cofactor formula for the inverse of a matrix (Dummit p365) shows
that X − 1 also has coeCcients 3 in R . In more detail, if X is the transpose of the matrix of
cofactors of X ,then X
×
I ,andso(det X ) − 1 X is the inverse of X . It follows
that the set GL n ( R ) of such matrices is a group. For example GL n (
·
X =det X
·
Z
) is the group of all
n × n matrices with integer coeCcients and determinant ± 1.
(c) If G and H are groups, then we can construct a new group G × H , called the product
of G and H . As a set, it is the Cartesian product of G and H , and multiplication is defined
by:
( g,h )( g ,h )=( gg ,hh ) .
(d) A group is commutative (or abelian )if
ab = ba, all a,b
G.
Recall from Math 593 the following classification of finite abelian groups. Every finite abelian
group is a product of cyclic groups. If gcd( m,n )=1,then C m
∈
×
C n contains an element of
C mn , and isomorphisms of this type give the only ambiguities
in the decomposition of a group into a product of cyclic groups.
From this one finds that every finite abelian group is isomorphicto exactly one group of
the following form:
×
C n
≈
C n 1 ×···×
C n r , n 1
|
n 2 ,...,n r− 1
|
n r .
2 This means, in particular, that R has an identity element 1. Homomorphisms of rings are required to
take 1 to 1.
3 This also follows from the Cayley-Hamilton theorem.
In both cases C m
order mn ,andso C m
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GROUP THEORY
3
n r .
Alternatively, every abelian group of finite order m is a product of p -groups, where p
ranges over the primes dividing m ,
G
≈
G p .
p|m
For each partition
1 ,
of n , there is a group C p n i of order p n , and every group of order p n is isomorphicto exactly
one group of this form.
(e) Permutation groups. Let S be a set and let G the set Sym( S )ofbijections α : S
n = n 1 +
···
+ n s , n i
≥
→
S .
group on n letters is S n =Sym( { 1 ,...,n} ), which has order n !. The symbol 1234567
◦
β . For example, the permutation
2574316
denotes the permutation sending 1
→
2, 2
→
5, 3
→
7, etc..
1.2. Subgroups.
Proposition 1.4. Let G be a group and let S be a nonempty subset of G such that
(a) a,b ∈ S = ⇒ ab ∈ S.
(b) a ∈ S = ⇒ a − 1
∈ S.
Then the law of composition on G makes S into a group.
Proof. Condition (a) implies that the law of composition on G does define a law of compo-
sition S × S → S on S . By assumption S contains at least one element a , its inverse a − 1 ,
and the product e = aa − 1 . Finally (b) shows that inverses exist in S .
S is
injective, and hence (by counting) bijective; in particular, 1 is in the image, and this implies
that a − 1
∈
S ,themap x
→
ax : S
→
∈
S . The example
N ⊂ Z
(additive groups) shows that (a) does not imply (b) when
G is infinite.
Proposition 1.5. An intersection of subgroups of G isasubgroupofG.
Proof. It is nonempty because it contains 1, and conditions (a) and (b) of the definition are
obvious.
Remark 1.6. It is generally true that an intersection of sub-algebraic-objects is a subobject.
For example, an intersection of subrings is a subring, an intersection of submodules is a
submodule, and so on.
Proposition 1.7. For any subset X of a group G, there is a smallest subgroup of G con-
taining X. It consists of all finite products (allowing repetitions) of elements of X and their
inverses.
Proof. The intersection S of all subgroups of G containing X is again a subgroup containing
X , and it is evidently the smallest such group. Clearly S contains with X , all finite products
of elements of X and their inverses. But the set of such products satisfies (a) and (b) of
(1.4) and hence is a subgroup containing X . It therefore equals S .
The order of this group is n 1 ···
Then G becomes a group with the composition law αβ = α
A subset S as in the proposition is called a subgroup of G .
If S is finite, then condition (a) implies (b): for any a
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