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Let $H$ and $K$ be two groups and suppose

  1. $H$ acts on $K$.
  2. $H$ preserves the group structure of $K$.
  3. $\phi$ is the permutation representation of the action:

$\phi:H\rightarrow Aut(K)$

$\phi(x)(k)=k^x$

Then the set $K\times H$ becomes a group with the product $(a,x)(b,y)=(ab^{x^{-1}},xy)$. This group is denoted by $K\rtimes_\phi H$.

Do two permutation isomorphic (group-structure-preserving) actions from $H$ on $K$, produce isomorphic groups $K\rtimes H$?

How about equivalent actions?


if $K\rtimes_\phi H$ and $K\rtimes_\psi H$ are isomorphic are the actions represented by $\phi$ and $\psi$ isomorphic?

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What do you mean by '2. $H$ preserves the group structure of $K$'? And, what is the difference between isomorphic and equivalent action of $H$ on $K$? –  Berci Jan 27 '13 at 22:04
    
preserves means: $(ab)^x=a^x b^x$ –  user59671 Jan 27 '13 at 22:06
    
2 actions from $H$ on $K$ are isomorphic iff there exist a bijection $\lambda:K\rightarrow K$ and an isomorphsim $\psi:H\rightarrow H$ such that $\lambda(k^x)=\lambda(k)^{\psi(x)}$. if $\psi$ is identity the actions are called equivalent. –  user59671 Jan 27 '13 at 22:09
    
Defintions come from books.google.com/… –  user59671 Jan 27 '13 at 22:34
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1 Answer

up vote 1 down vote accepted

Since you are not requiring your bijection $\lambda:K \to K$ to be a group isomorphism, the answer to your first two questions is no. I can think of a counterexample in which $K$ is cyclic of order 8 and $H$ has order 2.

The answer to the final question is also no. Let $G = K \times H$, with $K$ dihedral of order $2n$ for some $n>2$, and $H$ cyclic of order 2. By choosing a different complement of $K$ in $G$, we can also express $G$ as $K \rtimes H$ with nontrivial action of $H$ on $K$.

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