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
Definitions and
Examples of Groups
The Set of 1-1 mappings of
S onto itself, A(S)
Lemma 1. A(S) satisfies the following:
a) f, g A(S) implies that f ∘ g A(S).
b) f, g, h A(S) implies that (f ∘ g)∘h = f ∘ (g ∘ h)
c) There exists an element –the identity mapping, i –such that i∘f = f∘i = f for every f A(S).
1. If f Sn, show that there is some positive
integer k, depending on f, such that f k = i.
2. If S has three or more elements, show that we can find f, g A(S) such that fg gf.
3. For f A(S), let C(f) = {g A(S) | fg = gf}.
Prove that:
a) g, h C(f) implies that gh C(f).
b) g C(f) implies that g -1 C(f).
c) C(f) is not empty.
Definitions and Examples of
Groups
Definition. A nonempty set G is said to be a group if in G there is defined an operation such that:
Group Not a Group
(R, +) (R, •)
(Z, +) (N, +)
(Q, +) (Q, •)
(R\{0}, •) (R\{0}, +)
({0}, -) (Z, -)
(R+, •) (R-, •)
(all polynomials, +) (all polynomials, •) (Q \{0}, •) (Q \{0}, +)
Finite Groups
A group G is said to be a finite group if it has a finite number of elements. The number of elements in G is called the order of G and is denoted by |G|.
Let a be an element of a group G. If there is a smallest positive integer n such that an = e
Abelian Groups
A group
G
is said to be
abelian
if
a
b =
b
a
for all
a
,
b
G
.
The word abelian derives from the name
of the great Norwegian mathematician
Examples (Nonabelian)
1) Let G be the set of all mappings Ta,b:
defined by Ta,b(x) = ax + b for any real
number x, where a, b are real numbers and
a 0. Under the composition, G forms a
nonabelian group. Verify the following formula
Ta,bTc,d = Tac,ad+b
2. Let G = {f A(S) | f(s) s for only a finite
number of s S}, where S is supposed to
be an infinite set. Under the product in A(S),
Simple Remarks
Lemma 4
. If G is a group, then:
a) Its identity element is
unique
.
b) Every
a
G
has a
unique
inverse
a
-1
G
.
c) If
a
G,
(
a
-1)
-1=
a
.
Proof of Lemma 2 (b)
Let
G
be a group and
a
G
. Let
e
be the
identity of
G
. Suppose
b
,
c
G
are both
inverses of
a
. Then
ab
=
e
and
ac
=
e
. So
For each of the rules 1-5, either prove that the rule is true in any group, or give a counterexample:
1. If x2 = e then x = e.
2. If x2 = a2 then x = a.
3. (ab)2 = a2b2
4. If x2 = x then x = e.
5. For every x G there is y G such that x = y2.
6. Assume that a, b commute in G, i.e., ab = ba. Prove: a-1 commutes with b-1; a commutes
with ab; a2 commutes with b2.
7. Find all multiplication tables of groups of order 4 with headlines and sidelines labeled 1, a, b,
c. Hint: You can assume that 1 is the neutral element and a has order 2. You will then find only 2 possible multiplication tables.
8. If G is a group in which a2 = e for all a G, show that G is abelian.
9. Show that a group of order 5 must be abelian.
10. If G is a finite group, prove that, given a
G, there is an integer n > 0, depending on
a, such that an = e.
11. In Problem 10, show that there is an integer m > 0 such that am = e for all a G.
Question?
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please ask questions =(^ y ^)=
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