Understanding Semidirect Products: Homomorphisms and Group Structures Explained

[murmillo]

1. Homework Statement [/b]
I'm reading about semidirect products, and I don't understand this part:
Given two abstract groups H and K and a homomorphism
f : K --> AutH, define a group structure on the Cartesian product H X K
by the rule
(h₁, k₁) * (h₂; k₂) = (h₁ x f(k₁)(h₂), k₁k₂).
I don't understand how how h₁ x f(k₁)(h₂) is an element of H.


3. The Attempt at a Solution [/b]
I think that h₁ x f(k₁)(h₂) is an element of H only when H is normal. But the rule is supposed to work for any two groups H and K.

 

[micromass]

h₁ is an element of H.
f(k₁) is an automorphism of H, thus it takes elements of H to elements of H. In particular f(k₁)(h₂) is an element of H.
Multiplying the two element gives an element of H.