Garis-garis Besar
Garis-garis Besar
Perkuliahan
Perkuliahan
15/2/10 Sets and Relations
22/2/10 Definitions and Examples of Groups 01/2/10 Subgroups
08/3/10 Lagrange’s Theorem
15/3/10 Mid-test 1
22/3/10 Homomorphisms and Normal Subgroups 1 29/3/10 Homomorphisms and Normal Subgroups 2 05/4/10 Factor Groups 1
12/4/10 Factor Groups 2
19/4/10 Mid-test 2
26/4/10 Cauchy’s Theorem 1 03/5/10 Cauchy’s Theorem 2 10/5/10 The Symmetric Group 1 17/5/10 The Symmetric Group 2
Homomorphisms
Homomorphisms
and Normal
and Normal
Subgroups
Homomorphisms
Homomorphisms
Definition
Definition. Let G, G’ be two groups; then the mapping : G G’ is a homomorphismhomomorphism if
(ab) = (a)(b) for all a, b G.
The product on the left side—in (ab)—is that of G,
while the product (a)(b) is that of G’.
Examples
Examples
1. Let G be the group of all positive reals under the
multiplication of reals, and let G’ the group of all reals under addition. Let : G G’ be defined by
(x) = log10(x) for x G.
2. Let G be an abelian group and let : G G be
defined by (x) = x2.
3. Let G be the group of integers under + and G’ = {1,
Homomorphisms
Isomorphic Groups
Isomorphic Groups
Two groups
G
and
G
’ are said to be
isomorphic
isomorphic
if there is an
isomorphism of
G
onto
G
’.
Examples
Examples
4. Let G be any group and let A(G) be the set of all 1-1
mappings of G onto itself—here we are viewing G
merely as a set, forgetting about its multiplication.
• Given a G, define Ta : G G by
Ta(x) = ax for every x G.
Verify that Ta Tb = Tab.
• Define : G A(G) by (a) = Ta for every a G.
Cayley’s Theorem
Cayley’s Theorem
Theorem 1
Theorem 1. Every group G is isomorphic to
some subgroup of A(S), for an appropriate S.
Arthur Cayley
Arthur Cayley (1821-1895) was an English
mathematician who worked in matrix theory, invariant theory, and many other parts of
Homomorphism Properties
Homomorphism Properties
Lemma 1
Lemma 1. If is a homomorphism of G
into G’, then:
a) (e) = e’, the identity element of G’.
Image and Kernel
Image and Kernel
Definitions
Definitions. If is a homomorphism of G
into G’, then:
a) the imageimage of , (G), is defined by
(G) = {(a) | a G}.
b) the kernelkernel of , Ker , is defined by
Image and Kernel
Image and Kernel
Lemma 2
Lemma 2. If is a homomorphism of G
into G’, then:
a) the imageimage of is a subgroup of G’.
b) the kernelkernel of is a subgroup of G.
c) if w’ G’ is of the form (x) = w’, then
Kernel
Kernel
Theorem 2
Theorem 2. If is a homomorphism of G
into G’, then:
a) Given a G, a-1(Ker )a Ker .
b) is monomorphism if and only if
Normal Subgroups
Normal Subgroups
Definition
Definition. A subgroup N of G is said to be
a normal subgroupnormal subgroup of G if a-1Na N for
every a G.
We write “N is a normal subgroup of G” as
N
G.Theorem 3
Theorem 3. N
G if and only if every leftExamples
Examples
1. In Example 8 of Section 1, H = {Ta,b | a
rational}
G.2. The center Z(G) of any group G is a
normal subgroup of G.
3. In Section 1, the subgroup N = {i, f, f2} is
Problems
Problems
1. Let G be any group and A(G) the set of all 1-1
mappings of G, as a set, onto itself. Given a in G, define La : G G by La(x) = xa-1. Prove that:
a) La A(G)
b) LaLb = Lab
c) The mapping : G A(G) defined by (a) =
Problems
Problems
3. An automorphismautomorphism of G is an isomorphism from G to G itself. A
subgroup T of a group G is called characteristiccharacteristic if (T) T for all
automorphisms, , of G. Prove that:
a) M characteristic in G implies that M G.
b) M, N characteristic in G implies that MN is characteristic
in G.
c) A normal subgroup of a group need not be characteristic.
Question?
Question?
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If you are confused like this kitty is,
please ask questions =(^ y ^)=please ask questions =(^ y ^)=
If you are confused like this kitty is,
If you are confused like this kitty is,
