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Garis-garis Besar 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
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Garis-garis Besar 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 22/5/10 Final-exam
Homomorphisms Definition. Let G, G’ be two groups; then the mapping : G G’ is a homomorphism 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’. A homomorphism preserves the operation of G.
Examples • 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. • Let G be an abelian group and let : G G be defined by (x) = x2. • Let G be the group of integers under + and G’ = {1, -1}, the subgroup of the reals under multiplication. Define (m) = 1 if m is even, (m) = -1 if m is odd.
Homomorphisms • A homomorphism : G G’ is called monomorphism if a, b G: a b (a) (b). • A homomorphism : G G’ is called epimorphism if a’ G’: a G (a) = a’. • A homomorphism : G G’ is called isomorphism if it is both 1-1 and onto.
Isomorphic Groups Two groups G and G’ are said to be isomorphic if there is an isomorphism of G onto G’. We shall denote that G and G’ are isomorphic by writing G G’.
Examples • 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. Verify that is a monomorphism.
Cayley’s Theorem Theorem 1. Every group G is isomorphic to some subgroup of A(S), for an appropriate S. Arthur Cayley (1821-1895) was an English mathematician who worked in matrix theory, invariant theory, and many other parts of algebra.
Homomorphism Properties Lemma 1. If is a homomorphism of G into G’, then: • (e) = e’, the identity element of G’. • (a-1) = (a)-1 for all a G.
Image and Kernel Definitions. If is a homomorphism of G into G’, then: • the image of , (G), is defined by (G) = {(a) | a G}. • the kernel of , Ker , is defined by Ker = {a | (a) = e’}.
Image and Kernel Lemma 2. If is a homomorphism of G into G’, then: • the image of is a subgroup of G’. • the kernel of is a subgroup of G. • if w’ G’ is of the form (x) = w’, then -1(w’) is the coset (Ker ) x.
Kernel Theorem 2. If is a homomorphism of G into G’, then: • Given a G, a-1(Ker )a Ker . • is monomorphism if and only if Ker = (e).
Normal Subgroups Definition. A subgroup N of G is said to be a normal subgroup of G if a-1Na N for every a G. We write “N is a normal subgroup of G” as NG. Theorem 3. NG if and only if every left coset of N in G is a right coset of N in G.
Examples • In Example 8 of Section 1, H = {Ta,b | a rational} G. • The center Z(G) of any group G is a normal subgroup of G. • In Section 1, the subgroup N = {i, f, f2} is a normal subgroup of S3.
Problems • 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: • La A(G) • LaLb = Lab • The mapping : G A(G) defined by (a) = La is a homomorphism of G into A(G).
Problems • An automorphism of G is an isomorphism from G to G itself. A subgroup T of a group G is called characteristic if (T) T for all automorphisms, , of G. Prove that: • M characteristic in G implies that M G. • M, N characteristic in G implies that MN is characteristic in G. • A normal subgroup of a group need not be characteristic. • If N G and H is a subgroup of G, show that HN H.
Problems • If G is a nonabelian group of order 6, prove that G S3. • Let G be a group and H a subgroup of G. Let S be the set of all right cosets of H in G. Define, for b G, Tb : S S by Tb(Ha) = Hab-1. • Prove that TbTc = Tbc for all b, c G [then defines a homomorphism of G into A(S)]. • Describe Ker , the kernel of : G A(S). • Show that Ker is the largest normal subgroup of G lying in H [largest in the sense that if N G and N H, then N Ker .
Question? If you are confused like this kitty is, please ask questions =(^ y ^)=
