As a definition, if for a group $(G$|$\Omega)$; the orders of $G_{\omega}$ ($\omega$ in $\Omega$) are equal to eachother, then $G$ is said to be a $1/2$ -transitive group . Any example for such these groups? Thanks.
Searching for a $1/2$ -transitive group
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group-theory
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4What is $\Omega$ and what does $G_{\omega}$ mean? – 2012-04-30
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0Is $G_{\omega}$ the stabilizer of $\omega$, or the $G$-orbit of $\omega$? – 2012-04-30
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0Any Frobenius group which is not transitive. The nonabelian group of order 21 springs to mind... – 2012-04-30
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3But Frobenius groups are transitive by definition. Frobenius groups are examples of 3/2-transitive groups. i.e. their point stabilizers are 1/2-transitive. (Of course all transitive groups, including Frobenius groups, are also 1/2-transitive, but Babak Sorouh is presumably looking for examples that are 1/2-transitive but not transitive). The smallest such example is the trivial group acting on two points. – 2012-04-30
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1Arturo: $G_\omega$ normally means the stabilizer of $\omega$, but you would actually end up with an equivalent definition if you took it to mean the orbit of $\omega$. – 2012-04-30
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0@DerekHolt: yes I meant the point stabilizer. – 2012-05-01
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0I'm not sure why this merits a minus vote! – 2012-05-01
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0@DerekHolt: Thanks for answer. – 2012-05-01
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0@DerekHolt: Dear Prof. You meant"...looking for examples that are 1/2-transitive but not transitive..." above cause every transitive group acting on a set is clearly 1/2-transitive on that set? – 2012-05-02
1 Answers
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Here are a couple of examples of 1/2-transitive groups actions.
The cyclic group generated by any permutation in which all cycles have the same length, such as $(1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)$.
In your notation, if $(G_1|\Omega_1),\ldots,(G_k|\Omega_k)$ are group actions, then there is a natural action of the direct product $G_1 \times \cdots \times G_k$ on $\Omega_1 \cup \ldots \cup \Omega_k$. If each of the individual actions is transitive and all $|\Omega_i|$ are equal, then the resulting direct product action is 1/2-transitive.
For example, we could fix $n$ and let each $(G_i|\Omega_i)$ be the symmetric group in its natural action with $|\Omega_i|=n$. Then the direct product action is 1/2-transitive with degree $kn$.