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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

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Factor Groups

and

_and

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Theorem 1

If

N



G and

G/N = {Na | a  G},

then G/N is a group under the operation

(Na)(Nb) = Nab.

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Theorem 2

If

G is a finite abelian group of order |

G| and

p is a prime that divides |

G|, then G has

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Problems

1. If G is a cyclic group and N is a subgroup of G, show

that G/N is a cyclic group.

2. If G is an abellian group and N is a subgroup of G,

show that G/N is an abelian group.

3. Let G be an abelian group of order mn, where m and n

are relatively prime. Let M = {a  G | am = e}. Prove

that:

 M is a subgroup of G.

 G/M has no element, x, other than the identity element,

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Theorem 3

First Homomorphism Theorem First Homomorphism Theorem

Let  be a homomorphism of G onto G’ with kernel

K. Then G’  G/K, the isomorphism between

these being effected by the map

 : G/K  G’

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Theorem 4

Correspondence Theorem Correspondence Theorem

Let  be a homomorphism of G onto G’ with kernel

K. If H’ is a subgroup of G’ and if

H = {a  G | (a)  H’},

then H is a subgroup of G, K  H, and H/K  H’.

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Theorem 5

Second Homomorphism Theorem Second Homomorphism Theorem

Let H be a subgroup of a group G and N a normal

subgroup of G. Then HN = {hn| h  H, n  N}

is a subgroup of G, HN is a normal subgroup of

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Theorem 6

Third Homomorphism Theorem

If  is a homomorphism of G onto G’ with

kernel K, then, if N’  G’ and

N = {a  G | (a)  N’},

we conclude that G/N  G’/N’. Equivalently,

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Problems

1. Let G be the group of all real-valued functions on the unit interval

[0,1], where we define, for f, g  G, addition by (f+g)(x) = f(x)+g(x)

for every x [0,1]. If N = {f  G|f()=0}, prove that G/N  real

numbers under +.

2. If G₁, G₂ are two groups and G = G₁  G₂= {(a,b)|a  G₁, b  G₂},

where we define (a,b)(c,d) = (ac,bd), show that:

a) N = {(a,e₂)|a  G₁}, where e₂ is the unit element of G₂, is a

normal subgroup of G.

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Cauchy’s Theorem

Orbit Orbit

Let S be a set, f  A(S), and define a relation on S

as follows: s  t if t = f i (s) for some integer i. Verify

that this defines an equivalence relation on S.

The equivalence class of s, [s], is called the orbit

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Cauchy’s Theorem

Lemma 7

If f  A(S) is of order p, p a prime, then the

orbit of any element of S under f has 1 or p

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Cauchy’s Theorem

Theorem 8

If p is a prime and p divides the order of G,

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Cauchy’s Theorem

Lemma 9

Let

G

be a group of order

pq

, where

p,q

are

primes and

p > q.

If

a



G

is of order

p

and

A

is

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Cauchy’s Theorem

Corollary 10

If

G,

a

are as in Lemma 9 and

x



G

, then

x

-1

ax = a

i

, for some

i

where 0 <

i

<

p

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Cauchy’s Theorem

Lemma 11

If

a



G

is of order

m

and

b



G

is of order

n

, where

m

and

n

are relatively prime and

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Cauchy’s Theorem

Theorem 12

Let

G

be a group of order

pq,

where

p,q

are

primes and

p > q.

If

q



p

- 1, then G must

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Problems

1. Prove that a group of order 35 is cyclic.

2. Construct a nonabelian group of order 21. (Hint:

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Question?

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