[ Pobierz całość w formacie PDF ]
G/N is the collection of all cosets. Then (Na) · (Nb) = N(a · b) is a well defined
multiplication (binary operation) on G/N, and with this multiplication, G/N is a
group. Its identity is N and (Na)-1 =(Na-1). Furthermore, if G is finite, o(G/N) =
o(G)/o(N).
Proof
Multiplication of elements in G/N is multiplication of subsets in G.
(Na) · (Nb) =N(aN)b = N(Na)b = N(a · b).
Once multiplication is well defined,
the group axioms are immediate.
Exercise
Write out the above theorem for G an additive abelian group.
Example
Suppose G = Z under +, n 1, and N = nZ. Zn, the group of
integers mod n is defined by Zn = Z/nZ. If a is an integer, the coset a + nZ is
denoted by [a]. Note that [a] +[b] =[a + b], -[a] =[-a], and [a] =[a + nl] for any
integer l. Any additive abelian group has a scalar multiplication over Z, and in this
case it is just [a]m =[am]. Note that [a] =[r] where r is the remainder of a divided
by n, and thus the distinct elements of Zn are [0], [1], ..., [n - 1]. Also Zn is cyclic
because each of [1] and [-1] = [n - 1] is a generator. We already know that if p is a
prime, any non-zero element of Zp is a generator, because Zp has p elements.
Theorem
If n1 and a is any integer, then [a] is a generator of Zn iff (a, n) =1.
Proof
The element [a] is a generator iff the subgroup generated by [a] contains
[1] iff ∃ an integer k such that [a]k = [1] iff ∃ integers k and l such that ak + nl =1.
Exercise
Show that a positive integer is divisible by 3 iff the sum of its digits is
divisible by 3. Note that [10] = [1] in Z3. (See the fifth exercise on page 18.)
Homomorphisms
Homomorphisms are functions between groups that commute with the group op-
erations. It follows that they honor identities and inverses. In this section we list
28
Groups
Chapter 2
the basic properties. Properties 11), 12), and 13) show the connections between coset
groups and homomorphisms, and should be considered as the cornerstones of abstract
algebra.
Definition
If G and Ḡ are multiplicative groups, a function f : G → Ḡ is a
homomorphism if, for all a, b ∈ G, f(a · b) =f(a) · f(b). On the left side, the group
operation is in G, while on the right side it is in Ḡ. The kernel of f is defined by
ker(f) = f-1(ē) = {a ∈ G : f(a) = ē}. In other words, the kernel is the set of
solutions to the equation f(x) = ē.
(If Ḡ is an additive group, ker(f) =f-1(0).)
Examples
The constant map f : G → Ḡ defined by f(a) = ē is a homomorphism.
If H is a subgroup of G, the inclusion i : H → G is a homomorphism. The function
f : Z → Z defined by f(t) = 2t is a homomorphism of additive groups, while the
function defined by f(t) =t + 2 is not a homomorphism. The function h : Z → R - 0
defined by h(t) =2t is a homomorphism from an additive group to a multiplicative
group.
We now catalog the basic properties of homomorphisms. These will be helpful
later on when we study ring homomorphisms and module homomorphisms.
Theorem
Suppose G and Ḡ are groups and f : G → Ḡ is a homomorphism.
1)
f(e) = ē.
2)
f(a-1) =f(a)-1.
3)
f is injective ⇔ ker(f) =e.
4)
If H is a subgroup of G, f(H) is a subgroup of Ḡ. In particular, image(f) is
a subgroup of Ḡ.
5)
If H is a subgroup of Ḡ, f-1(H) is a subgroup of G. Furthermore, if H is
normal in Ḡ, then f-1(H) is normal in G.
6)
The kernel of f is a normal subgroup of G.
7)
If ḡ ∈ Ḡ, f-1(ḡ) is void or is a coset of ker(f), i.e., if f(g) = ḡ then
f-1(ḡ) =Ng where N= ker(f). In other words, if the equation f(x) = ḡ has a
¯
¯
¯
¯
¯
differential equations.
G
=
homomorphism, then h ◦ f : G →
is a homomorphism.
G = Ḡ, f is also called an automorphism.
Chapter 2
Groups
29
solution, then the set of all solutions is a coset of N= ker(f). This is a key fact
which is used routinely in topics such as systems of equations and linear
8)
9)
10)
11)
12)
13)
f
G
π
f
G/H
Ḡ
Thus defining a homomorphism on a quotient group is the same as defining a
homomorphism on the numerator which sends the denominator to ē. The
¯
¯
f is injective, and thus G/H ≈ image(f).
Given any group homomorphism f, domain(f)/ker(f) ≈ image(f). This is
the fundamental connection between quotient groups and homomorphisms.
=
The composition of homomorphisms is a homomorphism, i.e., if h : Ḡ →
is a
G
If f : G → Ḡ is a bijection, then the function f-1 : Ḡ → G is a homomorphism.
In this case, f is called an isomorphism, and we write G ≈ Ḡ. In the case
Isomorphisms preserve all algebraic properties. For example, if f is an
isomorphism and H ⊂ G is a subset, then H is a subgroup of G
iff f(H) is a subgroup of Ḡ, H is normal in G iff f(H) is normal in Ḡ, G is
cyclic iff Ḡ is cyclic, etc. Of course, this is somewhat of a cop-out, because an
algebraic property is one that, by definition, is preserved under isomorphisms.
Suppose H is a normal subgroup of G. Then π : G → G/H defined by
π(a) =Ha is a surjective homomorphism with kernel H. Furthermore, if
f : G → Ḡ is a surjective homomorphism with kernel H, then G/H ≈ Ḡ
(see below).
Suppose H is a normal subgroup of G. If H ⊂ ker(f), then f : G/H → Ḡ
¯
¯
defined by f(Ha) =f(a) is a well-defined homomorphism making
the following diagram commute.
¯
image of f is the image of f and the kernel of f is ker(f)/H. Thus if H = ker(f),
¯
30
Groups
Chapter 2
14) Suppose K is a group. Then K is an infinite cycle group iff K is isomorphic to
the integers under addition, i.e., K ≈ Z. K is a cyclic group of order n iff
K ≈ Zn.
Proof of 14)
[ Pobierz całość w formacie PDF ]