Download Lecture 10

Essential Properties of Homomorphisms
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Proposition 14: Let (G, ) and (H, ) be groups, and let
: G  H be a homomorphism. Then:
a)  (e) = e’ and  (x – 1) =  (x) – 1 for all x  G.
b) Im () is a subgroup of H.
c) Ker () is a normal subgroup of G.
d)  is injective if and only if Ker () = {e}
e) Furthermore, if Im () is a finite group, then
[G: Ker ()] = | Im () |.
Proposition 15: An  Sn for n  2.
Direct Product of Groups
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Definition: Let G and H be groups, not necessarily distinct. Then
the cartesian product GH with the composition (g1,h1) (g2,h2) =
(g1g2, h1h2) is a group, called the external direct product of G and
H, notation GH.
Remark 1: GH  HG
Remark 2: We can easily extend this idea to define the external
direct product of n groups, n  2.
Remark 3: If Gi, i = 1,2,….,n are finite groups, then:
|G1G2…….. Gn| = |G1| |G2|…….. |Gn|
Remark 4: The external direct product of abelian groups is abelian.
Example 1: Z2  Z2  K4.
Example 2: Extending the above idea, we get three abelian groups
of order 8: Z8 , Z4  Z2 , and Z2  Z2  Z2 . It can be shown that
these are the only abelian groups of order 8.
Internal Direct Product
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Definition: We define an operation on subsets of a group
by XY = {xy: x  X, y  Y}. It can easily be seen that
this operation is associative, i.e. (XY)Z = X(YZ).
Proposition 16: Let K and N be normal subgroups of a
group G such that (i) G = KN and (ii) K  N = {e}. Then:
a) Any element g  G is uniquely expressible in the form
g = kn, where k  K and n  N.
b) If g1 = k1n1 and g2 = k2n2 , then g1 g2 = (k1k2)(n1n2)
c) G  KN
• Definition: In this situation, we say that G is the internal
direct product of K and N.
Further Properties of Set Multiplication
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Proposition 17: Let G be a group, H a subgroup of G, and
K a normal subgroup of G. Then H  K is a normal
subgroup of H.
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Proposition 18: Let H and K be subgroups of a group G.
Then
HK is a subgroup of G if and only if HK = KH.
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Proposition 19: If H, K are finite subgroups of a group G,
then
|HK| = |H| |K| / | H  K |.
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Remark: Proposition 19 holds even if HK is not a
subgroup of G.
Quotient Structures
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Notation: Let G be a group, H a subgroup of G. Then we define
G/H = {xH: x  G}, i.e. the set of all left cosets of G.
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Proposition 20: Let K be a normal subgroup of the group G. Then
G/K is a group with respect to the operation of set multiplication
defined earlier, and furthermore (xK)(yK) = (xy)K for all x, y  G.
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Definition: The group G/K, where K is a normal subgroup of G, is
known as the quotient group. In case K has finite index in G, then
|G/K| = [G:K] and in case G is a finite group, then |G/K| = |G|/|K|.
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Proposition 21 (Fundamental Theorem on Homomorphisms):
a) If K is a normal subgroup of a group G then the map : G  G/K
given by (x) = xK for all x  G is a surjective homomorphism,
known as the natural homomorphism.
b) Let : G  H be a group homomorphism and let K = Ker ( ).
Then the map ’: G/K  Im () given by ’(xK) =  (x) is a group
isomorphism.