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Garis-garis Besar
Perkuliahan
15/2/10
Sets and Relations
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

Definitions and Examples of Groups
Subgroups

Lagrange’s Theorem
Mid-test 1
Homomorphisms and Normal Subgroups 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

Factor Groups and
Cauchy’s Theorem

Theorem 1
If N  G and

G/N = {Na | a  G},

then G/N is a group under the operation
(Na)(Nb) = Nab.
If G is a finite group and N  G, then |G/N| =

|G|/|N|.

Theorem 2
If G is a finite abelian group of order |G| and

p is a prime that divides |G|, then G has
an element of order p.

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,
such that xm = unit element of G/M.

Theorem 3
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’
defined by (Ka) = (a).

Theorem 4
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’.
Finally, if H’  G’, then H  G.

Theorem 5
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
H, and H/(HN)  (HN)/N.

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,
G/N  (G/K)/(N/K).

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.

b)

N  G1.

c)

G/N  G2.

Cauchy’s Theorem
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
of s under f.

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
elements.

Cauchy’s Theorem
Theorem 8
If p is a prime and p divides the order of G,
then G contains an element of order p.

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
the subgroup of G generated by a, then A  G.

Cauchy’s Theorem
Corollary 10
If G, a are as in Lemma 9 and x  G, then
x-1ax = ai, for some i where 0 < i < p

(depending on x)

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
ab = ba, c = ab is of order mn.

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
be cyclic.

Problems
1.

Prove that a group of order 35 is cyclic.

2.

Construct a nonabelian group of order 21. (Hint:
Assume that a3 = e, b7 = e and find some i such that

a-1ba = ai ≠ a, which is consistent with the relations
a3 = b7 = e.)
3.

Let G be a group of order pnm, where p is prime and
p  m. Supposse that G has a normal subgroup of order
pn. Prove that (P) = P for every automorphism  of G.

Question?

If you are confused like this kitty is,
please ask questions =(^ y ^)=