Groups generated by derangements
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 group-theoretic structure of G/D(G); in particular, we construct groups G in which G/D(G) is not a Frobenius complement.
Citation
Bailey , R A , Cameron , P J , Giudici , M & Royle , G F 2021 , ' Groups generated by derangements ' , Journal of Algebra , vol. 572 , pp. 245-262 . https://doi.org/10.1016/j.jalgebra.2020.12.020
Publication
Journal of Algebra
Status
Peer reviewed
ISSN
0021-8693Type
Journal article
Description
Funding: the research of the last two authors is supported by the Australian Research Council Discovery Project DP200101951. This work was supported by EPSRC grant no EP/R014604/1. In addition, the second author was supported by a Simons Fellowship.Collections
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