Subdegree growth rates of infinite primitive permutation groups

Smith, Simon (2010) Subdegree growth rates of infinite primitive permutation groups. Journal of the London Mathematical Society, 82 (2). pp. 526-548. ISSN 0024-6107

Full content URL: https://doi.org/10.1112/jlms/jdq046

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Subdegree growth rates of infinite primitive permutation groups
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Abstract

A transitive group G of permutations of a set Ω is primitive if the only G-invariant equivalence relations on Ω are the trivial and universal relations. If α ∈ Ω, then the orbits of the stabilizer G_α on Ω are called the α-suborbits of G; when G acts transitively the cardinalities of these α-suborbits are the subdegrees of G. If G acts primitively on an infinite set Ω, and all the suborbits of G are finite, Adeleke and Neumann asked if, after enumerating the subdegrees of G as a non-decreasing sequence 1 = m_0 ≤ m_1 ≤ . . , the subdegree growth rates of infinite primitive groups that act distance-transitively on locally finite distance-transitive graphs are extremal, and conjecture that there might exist a number c that perhaps depends upon G, perhaps only on the size of some suborbit m, such that m_r ≤ c(m − 2)^{r−1}. In this paper it is shown that such an enumeration is not desirable, as there exist infinite primitive permutation groups possessing no infinite subdegree, in which two distinct subdegrees are each equal to the cardinality of infinitely many α-suborbits. The examples used to show this provide several novel methods for constructing infinite primitive graphs. A revised enumeration method is then proposed, and it is shown that, under this, Adeleke and Neumann’s question may be answered, at least for groups exhibiting suitable rates of growth.

Keywords:Primitive, Permutation groups
Subjects:G Mathematical and Computer Sciences > G110 Pure Mathematics
Divisions:College of Science > School of Mathematics and Physics
ID Code:46549
Deposited On:10 Nov 2021 10:42

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