scholarly journals Derangements in cosets of primitive permutation groups

2015 ◽  
Vol 18 (1) ◽  
Author(s):  
Andrei Pavelescu
Keyword(s):  
Fixed Point ◽  
Finite Fields ◽  
Permutation Groups ◽  
Normal Subgroups ◽  
Primitive Groups ◽  
Branched Coverings ◽  
Affine Groups ◽  
Nonabelian Subgroup ◽  
Projective Curves ◽  
Curves Over Finite Fields

AbstractMotivated by questions arising in connection with branched coverings of connected smooth projective curves over finite fields, we study the proportion of fixed-point free elements (derangements) in cosets of normal subgroups of primitive permutations groups. Using the Aschbacher–O'Nan–Scott Theorem for primitive groups to partition the problem, we provide complete answers for affine groups and groups which contain a regular normal nonabelian subgroup.

Bounds for finite semiprimitive permutation groups: order, base size, and minimal degree

2020 ◽  
Vol 63 (4) ◽  
pp. 1071-1091
Author(s):  
Luke Morgan ◽  
Cheryl E. Praeger ◽  
Kyle Rosa
Keyword(s):  
Fixed Point ◽  
Normal Subgroup ◽  
Symmetric Group ◽  
Minimal Normal Subgroup ◽  
Structural Information ◽  
Minimal Degree ◽  
Permutation Groups ◽  
Primitive Groups ◽  
Base Size ◽  
Abstract Structure

AbstractIn this paper, we study finite semiprimitive permutation groups, that is, groups in which each normal subgroup is transitive or semiregular. These groups have recently been investigated in terms of their abstract structure, in a similar way to the O'Nan–Scott Theorem for primitive groups. Our goal here is to explore aspects of such groups which may be useful in place of precise structural information. We give bounds on the order, base size, minimal degree, fixed point ratio, and chief length of an arbitrary finite semiprimitive group in terms of its degree. To establish these bounds, we study the structure of a finite semiprimitive group that induces the alternating or symmetric group on the set of orbits of an intransitive minimal normal subgroup.

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