Let G and H be groups and H Aut Ga homomorphism Let G X H be the set G X H with the following binary operation (g,h)(g ',h ') (g (h)(g ') ,hh ') Show that G X H is a group with identity element (e,e) and (g,h) 1 ((h 1 )(g 1 ),h 1 ) G X H is called the semidirect product of G and H ...

To show that G X H is a group we need to show that it satisfies the four group axioms closure asso ...

Let G and H be groups and : H Aut Ga homomorphism. Let G X

Question:

Let G and H be groups and θ : H → Aut Ga homomorphism. Let G X_θ H be the set G X H with the following binary operation: (g,h)(g',h') = (g[θ(h)(g')],hh'). Show that G X_θH is a group with identity element (e,e) and (g,h)^-1 = (θ(h^-1)(g^-1),h^-1). G X_θH is called the semidirect product of G and H.