Page 95 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes =  (g) x[ (g)] -1
-1
-1
(x). (Note that -1(g)  G.)
= f  1 (g )

-1  Inn G    Aut G and f  Inn G.
g
  o f o  = f  1 (g) g

 Inn G  Aut G.
Now we will prove an interesting result which relates the cosets of the centre of a group G to
lnn G. Recall that the centre of G, Z(G) = { x  G | xg = gx  g  G }.
Theorem 7: Let G be a group. Then G/Z(G)  Inn G.

Proof: As usual, we will use the powerful Fundamental Theorem of Homomorphism to prove
this result.
We define f : G  Aut G : f(g) = f .
g
Firstly, f is a homomorphism because for g, h  G,
f(gh) = f gh
= I, o f (sec proof of Theorem 13)
h
= [(g) o f(h).

Next, Im F = ( f , 1 g  G ) = Inn G.
g
Finally, Ker f = ( g  G | f, = I }
G
= { g  G [ f (x) = x  x G }
g
= { g G | gxg = x  x G }
-1
= { g G | gx = xg  x G }
= Z(G).
Therefore, by the Fundamental Theorem,
G/Z(G)  Inn G.

Self Assessment

1. An isomorphism of a group G itself is called as an ................. of G.

(a) Homomorphism (b) automorphism
(c) Herf (d) one-to-one function
2. The word isomorphisms is derived from Greek word ISOS meaning .................
(a) equal (b) unequal

(c) bijective (d) subjective
7.3 Summary

The proof of the Fundamental Theorem of Homomorphism, which says that if f : G  G
 1 2
is a group homomorphism, then G /Ker f  Im f,
1

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