Page 94 - DMTH403_ABSTRACT_ALGEBRA

Unit 7: Homomorphism Theorem

We will show that f is an automorphism of G. Notes
g
(i) f, is a homomorphism : If x, y  G, then

f (xy) = g(xy) g -1
g
= gx(e) yg , where e is the identity of G.
-1
= gx(g g) yg -1
-1
= (gxg ) (gyg- )
1
-1
= f (x) f (dy).
g
g
(ii) f is 1-1 : For x, y  G, f (x) = f (y)  gxg = gyg  x = y, by the cancellation laws in G.
-1
-1
g
g
g
(iii) f is onto : If y  G, then
g
Y = (gg )y(gg )
-1
-1
= (g yg)g -1
-1
= f (g yg)  lm f .
-1
g
g
Thus, f, is an automorphism of G.
Definition: f is called an inner automorphism of G induced by the element g in G. The subset of
g
Aut G consisting of all inner automorphism of G is denoted by Inn G.
For example, Let us compute f (1). f (l 3) and f (1 2 3), where g = (1 2). Note that g = (1 2) = g.
-1
g
g
g
Now, f (1) = g a I o g = I,
-1
g
f (l 3) = (1 2) (1 3) (1 2) = (2 3).
g
f (l 2 3) = (1 2)(1 2 3)(1 2) = (1 3 2).
g
Theorem 6: Let G be a group. Then Inn G is a normal subgroup of Aut G. i
Proof: Inn G is non-empty, because I = f  Inn G, where e is the identity in G.
e
G
Now, let us see if f, o fh  Inn G for g, h  G.
For any x  G, f . f (x) = fg(hxh )
-1
g
n
= g(hxh ) g -1
-1
= (gh)x (gh) -1
= fgh(x)
Thus, f = f o f , i.e., Inn G is closed under composition. Also f = I belongs to Inn G.
gh
h
g
G
e
Now, for f  Inn G, 3 fg  Inn G such that
-1
g
f o f = f gg -1 = f = I . Similarly, f o f = I .
-1
g
g
G
g-1
e
g
G
Thus, f = (f ) . That is every element of Inn G has an inverse in Inn G.
-1
-1
g
g
This proves that Inn G is a subgroup of Aut G.
Now, to prove that Inn  Aut G, let   Aut G and f  Inn G. Then, for any x  G
g
 × f × (x) =  × f ( (x))
-1
-1
g
g
=  (g(x)g )
-1
-1
=  (g)  ((x))  (g )
-1
-1
-1
-1
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