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Garis-garis Besar Perkuliahan 15/2/10 22/2/10 01/2/10 08/3/10 15/3/10 22/3/10 29/3/10 05/4/10 12/4/10 19/4/10 26/4/10 03/5/10 10/5/10 17/5/10 22/5/10 Sets and Relations Definitions and Examples of Groups Subgroups Lagrange’s Theorem Mid-test 1 Homomorphisms and Normal Subgroups 2 Factor Groups 1 Factor Groups 2 Mid-test 2 Cauchy’s Theorem 1 Cauchy’s Theorem 2 The Symmetric Group 1 The Symmetric Group 2 Final-exam

Subgroups Section 2

Definition of a Subgroup A nonempty subset H of a group G is called a subgroup of G if, relative to the product in G, H itself forms a group. A = {1, -1} is a group under the multiplication of integers, but is not a subgroup of Z viewed as a group with respect to +.

Lemma 3 A nonempty subset H of a group G is subgroup if and only if H is closed with respect to the operation of G and, given a H, then a-1 H.

Examples 1. The set of all even integers is a subgroup of the group of integers under +. 2. Let m > 1 be any integer. The set Hm of all multiple of m in Z is a subgroup of Z under +. 3. Let a S and let H(a) = {f A(S) | f(a) = a}. Then H(a) is a subgroup of A(S). 4. Let G be any group and let a G. The set A = {ai | i any integer} is a subgroup of G.

Cyclic Subgroup The cyclic subgroup of G generated by a is a set {ai | i any integer}, denoted by (a). § If e is the identity element of G, then (e) = {e}. § § § U n = ( n) Z = (1) = (-1) Z 7* = (3) = (5)

More Examples Let G be any group. For a G: n The set C(a) = {g G | ag = ga} is a subgroup of G. It is called the centralizer of a in G. n The set Z(G) = {z G | xz = zx for all x G} is a subgroup of G. It is called the center of G.

Lemma 4 Suppose that G is a group and H is a nonempty finite subset of G closed under the product in G. Then H is a subgroup of G.

Corollary If G is a finite group and H is a nonempty subset of G closed under multiplication in G, then H is a subgroup of G.

Problems 1. Find all subgroups of S 3. 2. If G is cyclic, show that every subgroup of G is cyclic. 3. If G has no proper subgroups, prove that G is cyclic of order p, where p is a prime number.

Problems 4. If A, B are subgroups of an abelian group G, show that AB = {ab | a A, b B} is a subgroup of G. 5. Give an example of a group G and two subgroups A, B of G such that AB is not a subgroup of G. 6. Let G be a group, H a subgroup of G. Let Hx = {hx | h H}. Show that, given a, b G, then Ha = Hb or Ha Hb = .

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