april 18 2013 abstract algebra

session - 6 today, we will learn about: SUB GRUP 1. ( Zm, Xm)

2. permutation group

3. another examples of group

3. sub- group grup ?

sifat - sifat grup ? a x b = (a x b) mod m (Zm, Xm) group ? Theorem

The order of Sn is n!

If n = 3, then Sn is non-Abelian. permutation group Definition

A subset H of a group G is a subgroup of G if H itself a group with respect to the operation on G. review... m m Definition

A permutation of a nonempty set S is bijective mapping from S to S.. Theorem 6. 2 The set of all permutations of a nonempty set S is a group with respect to composition. Lemma 7. 2 Let (G,*) be a group, and H c G.

(a) If f is identity of H and e is identity of G,

then f = e.

(b) If a is in H, then the inverse of a in H is the same as the inverse of a in G. tidak sejati sejati sub grup G dan E, dengan E = {e} E < H < G order grup --> o(G) atau #G Teorema Lagrange:

Jika G grup hingga dan H subgrup dari G maka o(H) merupakan pembagi dari o(G) akibat :

untuk semua k bilangan bulat positif

dengan k|m maka himpunan <m/k> adalah subgrup dari Zm dengan order k Z adalah himpunan semua bilangan bulat modulo 36, sajikan semua subgrup dari Z 36 36 other examples of GROUP 2 "subgroup" theorems 1. "one step subgroup test" theorm.

2. "two steps subgroup test" theorm Let (G,*) be a group and H is a nonempty subset of G. H is a subgroup of G if:

(a) If a,b element of H, then a * b element of H, and

(b) If a element of H, then element invers of a is in H.

Let (G,*) be a group and H is a nonempty subset of G.

H is a subgroup of G if:

(a) H is nonempty, and

(b) If a,b element of H, then a * b is in H. one step two steps -1 grup siklis

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