In a context, the author wrote that for a finite permutation group $G$ acting on a set $\Omega$ of degree $n$, the list of subdegrees is an invariant of the group. What could the meaning of "invariant of $G$" be? Does it mean invariant of group under the group action or I am wrong?
What could the meaning of "invariant of $G$" be?
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$\begingroup$
terminology
finite-groups
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0What context? What is a subdegree? – 2012-05-23
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0@QiaochuYuan: Permutaion Groups by J.D.Dixon. It is a lenght of the orbit of one of the point stablizres. Thanks – 2012-05-23
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0I don't have this book and cannot understand what is meant here. Can you quote a longer passage? – 2012-05-23
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0@QiaochuYuan: Thanks for the time Prof.,in a Theorem 3.2B., he took somewhere a $\Sigma$ subset of $\Omega$ x $\Omega$ an $G$-invarant when for all $x\in G$, $\Sigma(\alpha)^x=\Sigma(\alpha^x)$ wherein a group $G$ acts transitively on the set $\Omega$. Is this the same he quoted above? – 2012-05-23
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0The lengths of the orbits of a point stabilizer are invariants of the *permutation group* $G$. So yes, it's an invariant of the group action. Different action have (generally) different subdegrees. – 2012-05-24