- #1

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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

(h

_{1}, k

_{1}) * (h

_{2}; k

_{2}) = (h

_{1}x f(k

_{1})(h

_{2}), k

_{1}k

_{2}).

I don't understand how how h

_{1}x f(k

_{1})(h

_{2}) is an element of H.

3. The Attempt at a Solution [/b]

I think that h

_{1}x f(k

_{1})(h

_{2}) is an element of H only when H is normal. But the rule is supposed to work for any two groups H and K.