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
LAGRANGE’S THEOREM
LAGRANGE’S THEOREM
EQUIVALENCE RELATION
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 (reflexivityreflexivity).
b) a ~ b implies that b ~ a (symmetrysymmetry).
EXAMPLES
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.
EQUIVALENCE CLASS
EQUIVALENCE CLASS
Definition
Definition. If ~ is an equivalence relation on S, then the classclass 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.
EQUIVALENCE CLASS
EQUIVALENCE CLASS
Theorem 1
Theorem 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, ~
LAGRANGE’S THEOREM
LAGRANGE’S THEOREM
Theorem 2
Theorem 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
ORDER OF AN ELEMENT
ORDER OF AN ELEMENT
Definition
Definition. If G is finite, then the order of a, written o(a), is the least positive integerleast positive integer m such that am = e.
Theorem 4
Theorem 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
CYCLIC GROUP
CYCLIC GROUP
A group G is said to be cycliccyclic if there is an element a G such that every element of G is a power of
a.
Theorem 3
CONGRUENCE CLASS MOD N
PROBLEMS
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 cosetsleft 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,
PROBLEMS
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.
PROBLEMS
PROBLEMS
6.
If
o(a) = m and as = e, prove that m | s.7. 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
QUESTION?
QUESTION?
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If you are confused like this kitty is,
