Download Lecture 11

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.
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 17: 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|.
Fundamental Theorem on
Homomorphisms
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Proposition 18 (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.