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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
1
29/3/10
2
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
Homomorphisms and Normal Subgroups
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
LAGRANGE’S THEOREM
SECTION 3
EQUIVALENCE RELATION
Definition.
Definition A relation ~ on a set S is called
an equivalence relation if, for all a, b, c S,
it satisfies:
a) a ~ a (reflexivity).
reflexivity
b) a ~ b implies that b ~ a (symmetry).
symmetry
c) a ~ b, b ~ c implies that a ~ c
(transitivity).
transitivity
EXAMPLES
1. Let n > 1 be a fixed integer. Define a ~ b
for a, b if n | (b – a). When a ~ b, we
write this as a b mod n, which is read
“a congruent to b mod n.”
2. Let G be a group and H a subgroup of G.
Define a ~ b for a, b G if ab-1 H.
3. Let G be any group. Declare that a ~ b if
there exists an x G such that b = x-1ax.
EQUIVALENCE CLASS
Definition.
Definition If ~ is an equivalence relation
on S, then the class of a, is defined by
[a] = {b S | b ~ a}.
In Example 2, b ~ a ba -1 H ba -1 = h
for some h H. That is, b ~ a b = ha
Ha = {ha | h H}. Thus, [a] = Ha.
The set Ha is called a right coset of H in G.
EQUIVALENCE CLASS
Theorem 1.
1 If ~ is an equivalence relation on
S, then S = [a], where this union runs over
one element from each class, and where [a]
[b] implies that [a] [b] = . That is, ~
partitions S into equivalence classes.
LAGRANGE’S THEOREM
Theorem 2.
2 If G is a finite group and H is a
subgroup of G, then the order of H
divides the order of G.
J. L. Lagrange (1736-1813) was a great
Italian mathematician who made
fundamental contributions to all the areas
of mathematics of his day.
ORDER OF AN ELEMENT
Definition.
Definition If G is finite, then the order of a, written
o(a), is the least positive integer m such that
am = e.
Theorem 4.
4 If G is finite and a G, then o(a) | |G|.
Corollary.
Corollary If G is a finite group of order n, then an
= e for all a G.
CYCLIC GROUP
A group G is said to be cyclic if there is an element
a G such that every element of G is a power of
a.
Theorem 3.
3 A group G of prime order is cyclic.
CONGRUENCE CLASS MOD N
Theorem 5. n forms a cyclic group under
addition modulo n.
Theorem 6. n* forms an abelian group
under the product modulo n, of order (n).
Theorem 7. If a is an integer relatively
prime to n, then a(n) 1 mod n.
Corollary (Fermat). If p is a prime and p a,
then
ap-1 1 mod p.
PROBLEMS
1. Let G be a group and H a subgroup of G.
Define a ~ b for a, b G if a-1b H. Prove
that this defines an equivalence relation on
G, and show that [a] = aH = {ah | h H}.
The sets aH are called left cosets of H in G.
2. If G is S3 and H = {i, f}, where f : S S is
defined by f(x1) = x2, f(x2) = x1, f(x3) = x3,
list all the right cosets of H in G and list all
the left cosets of H in G.
PROBLEMS
3. If p is a prime number, show that the only
solutions of x2 1 mod p are x 1 mod p
and x -1 mod p.
4. If G is a finite abelian group and a1, a2, an
are all its elements, show that x = a1a2an
must satisfy x2 = e.
5. If p is a prime number of the form 4n + 3,
show that we cannot solve x2 -1 mod p.
PROBLEMS
6. If
7.
o(a) = m and as = e, prove that m | s.
If in a group G, a5 = e and aba-1 = b2, find o(b) if
b e.
8. In a cyclic group of order n, show
that for each positive integer m that
divides n (including m = 1 and m =
n) there are (m) elements of order
m.
QUESTION?
If you are confused like this kitty is,
please ask questions =(^ y ^)=
PERKULIAHAN
15/2/10
22/2/10
01/2/10
08/3/10
15/3/10
22/3/10
1
29/3/10
2
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
Homomorphisms and Normal Subgroups
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
LAGRANGE’S THEOREM
SECTION 3
EQUIVALENCE RELATION
Definition.
Definition A relation ~ on a set S is called
an equivalence relation if, for all a, b, c S,
it satisfies:
a) a ~ a (reflexivity).
reflexivity
b) a ~ b implies that b ~ a (symmetry).
symmetry
c) a ~ b, b ~ c implies that a ~ c
(transitivity).
transitivity
EXAMPLES
1. Let n > 1 be a fixed integer. Define a ~ b
for a, b if n | (b – a). When a ~ b, we
write this as a b mod n, which is read
“a congruent to b mod n.”
2. Let G be a group and H a subgroup of G.
Define a ~ b for a, b G if ab-1 H.
3. Let G be any group. Declare that a ~ b if
there exists an x G such that b = x-1ax.
EQUIVALENCE CLASS
Definition.
Definition If ~ is an equivalence relation
on S, then the class of a, is defined by
[a] = {b S | b ~ a}.
In Example 2, b ~ a ba -1 H ba -1 = h
for some h H. That is, b ~ a b = ha
Ha = {ha | h H}. Thus, [a] = Ha.
The set Ha is called a right coset of H in G.
EQUIVALENCE CLASS
Theorem 1.
1 If ~ is an equivalence relation on
S, then S = [a], where this union runs over
one element from each class, and where [a]
[b] implies that [a] [b] = . That is, ~
partitions S into equivalence classes.
LAGRANGE’S THEOREM
Theorem 2.
2 If G is a finite group and H is a
subgroup of G, then the order of H
divides the order of G.
J. L. Lagrange (1736-1813) was a great
Italian mathematician who made
fundamental contributions to all the areas
of mathematics of his day.
ORDER OF AN ELEMENT
Definition.
Definition If G is finite, then the order of a, written
o(a), is the least positive integer m such that
am = e.
Theorem 4.
4 If G is finite and a G, then o(a) | |G|.
Corollary.
Corollary If G is a finite group of order n, then an
= e for all a G.
CYCLIC GROUP
A group G is said to be cyclic if there is an element
a G such that every element of G is a power of
a.
Theorem 3.
3 A group G of prime order is cyclic.
CONGRUENCE CLASS MOD N
Theorem 5. n forms a cyclic group under
addition modulo n.
Theorem 6. n* forms an abelian group
under the product modulo n, of order (n).
Theorem 7. If a is an integer relatively
prime to n, then a(n) 1 mod n.
Corollary (Fermat). If p is a prime and p a,
then
ap-1 1 mod p.
PROBLEMS
1. Let G be a group and H a subgroup of G.
Define a ~ b for a, b G if a-1b H. Prove
that this defines an equivalence relation on
G, and show that [a] = aH = {ah | h H}.
The sets aH are called left cosets of H in G.
2. If G is S3 and H = {i, f}, where f : S S is
defined by f(x1) = x2, f(x2) = x1, f(x3) = x3,
list all the right cosets of H in G and list all
the left cosets of H in G.
PROBLEMS
3. If p is a prime number, show that the only
solutions of x2 1 mod p are x 1 mod p
and x -1 mod p.
4. If G is a finite abelian group and a1, a2, an
are all its elements, show that x = a1a2an
must satisfy x2 = e.
5. If p is a prime number of the form 4n + 3,
show that we cannot solve x2 -1 mod p.
PROBLEMS
6. If
7.
o(a) = m and as = e, prove that m | s.
If in a group G, a5 = e and aba-1 = b2, find o(b) if
b e.
8. In a cyclic group of order n, show
that for each positive integer m that
divides n (including m = 1 and m =
n) there are (m) elements of order
m.
QUESTION?
If you are confused like this kitty is,
please ask questions =(^ y ^)=