Answer to Question #164064 in Abstract Algebra for K

Question

We showed that if S and T are isomorphic groups and e is the identity for S ,φ(e) gives the identity for T. Now show that if a and a′ are inverses in S, then φ (a) and φ (a′) are inverses in T

. (In other words, show that φ (a′) = (φ(a))'

Answer 1

Solution:

We have, a.a'=e. (since a and a' are inverse)

=> φ (a.a′) =φ (e) (applying φ to both sides)

=> φ (a).φ (a′) =φ (e) (as φ is isomorphism)

=> φ (a).φ (a′) =φ (a).[φ (a)]' (as φ(e) is identity of T)

=> φ (a′) =[φ (a)]' (left cancelling φ (a))

Hence proved.