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Semidirect product

  • Thread starter murmillo
  • Start date
  • #1
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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.
 

Answers and Replies

  • #2
22,097
3,280
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.
 

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