Suppose G G is a group homomorphism Prove that ghg1 ker for

Suppose : G G\' is a group homomorphism. Prove that ghg1 ker for all g G and h ker .

Solution

We know that the kernel is a subgroup..

Suppose that g G. We want to prove that g Ker g¹ Ker .

Suppose that h Ker . We need to prove that ghg¹ Ker .

Now (ghg¹ ) = (g) (h) g¹

= (g) f (g) ¹

= (g) (g) ¹

   = f. (= (e)), e the identity element)

Thus ghg¹ Ker