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Movement and Separation of Subsets of Points Under Group Actions
Author(s) -
Praeger Cheryl E.
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461079800578x
Subject(s) - combinatorics , bounded function , mathematics , group (periodic table) , permutation (music) , permutation group , invariant (physics) , finite group , orbit (dynamics) , element (criminal law) , discrete mathematics , mathematical analysis , physics , quantum mechanics , acoustics , law , political science , engineering , mathematical physics , aerospace engineering
Let G be a permutation group on a set Ω, and let m and k be integers where 0< m < k . For a subset Γ of Ω, if the cardinalities of the sets Γ g ∖Γ, for g ∈ G , are finite and bounded, then Γ is said to have bounded movement, and the movement of Γ is defined as move (Γ)=max g ∈ G ∣Γ g ∖Γ∣. If there is a k ‐element subset Γ such that move (Γ)⩽ m , it is shown that some G ‐orbit has length at most ( k 2 − m )/( k − m ). When combined with a result of P. M. Neumann, this result has the following consequence: if some infinite subset Γ has bounded movement at most m , then either Γ is a G ‐invariant subset with at most m points added or removed, or Γ nontrivially meets a G ‐orbit of length at most m 2 + m +1. Also, if move (Γ)⩽ m for all k ‐element subsets Γ and if G has no fixed points in Ω, then either ∣Ω∣⩽ k + m (and in this case all permutation groups on Ω have this property), or ∣Ω∣⩽5 m −2. These results generalise earlier results about the separation of finite sets under group actions by B. J. Birch, R. G. Burns, S. O. Macdonald and P. M. Neumann, and groups in which all subsets have bounded movement (by the author).