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Abstract
We examine the subgroup D(G) of a transitive permutation group G which is generated by the derangements in G. Our main results bound the index of this subgroup: we conjecture that, if G has degree n and is not a Frobenius group, then G:D(G)⩽n−1; we prove this except when G is a primitive affine group. For affine groups, we translate our conjecture into an equivalent form regarding H:R(H), where H is a linear group on a finite vector space and R(H) is the subgroup of H generated by elements having eigenvalue 1. If G is a Frobenius group, then D(G) is the Frobenius kernel, and so G/D(G) is isomorphic to a Frobenius complement. We give some examples where D(G)≠G, and examine the grouptheoretic structure of G/D(G); in particular, we construct groups G in which G/D(G) is not a Frobenius complement.
Original language  English 

Pages (fromto)  245262 
Number of pages  18 
Journal  Journal of Algebra 
Volume  572 
DOIs  
Publication status  Published  15 Apr 2021 
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 1 Finished

The synchronisation hierarchy of permutation groups
Bamberg, J., Giudici, M. & Royle, G.
ARC Australian Research Council
1/07/20 → 31/12/23
Project: Research