Groups generated by derangements

R. A.  Bailey; Peter J. Cameron; Michael Giudici; Gordon F. Royle

doi:10.1016/j.jalgebra.2020.12.020

Groups generated by derangements

R. A. Bailey, Peter J. Cameron, Michael Giudici, Gordon F. Royle

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	245-262
Journal	Journal of Algebra
Volume	572
Early online date	30 Dec 2020
DOIs	https://doi.org/10.1016/j.jalgebra.2020.12.020
Publication status	Published - 15 Apr 2021

Keywords

Permutation group
Derangement
Frobenius group
Linear group

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10.1016/j.jalgebra.2020.12.020

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Copyright © 2020 Elsevier Inc. All rights reserved. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1016/j.jalgebra.2020.12.020.
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@article{f57837ec428f4febb7481415297f7e6e,

title = "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.",

keywords = "Permutation group, Derangement, Frobenius group, Linear group",

author = "Bailey, {R. A.} and Cameron, {Peter J.} and Michael Giudici and Royle, {Gordon F.}",

note = "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.",

year = "2021",

month = apr,

day = "15",

doi = "10.1016/j.jalgebra.2020.12.020",

language = "English",

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pages = "245--262",

journal = "Journal of Algebra",

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T1 - Groups generated by derangements

AU - Bailey, R. A.

AU - Cameron, Peter J.

AU - Giudici, Michael

AU - Royle, Gordon F.

N1 - 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.

PY - 2021/4/15

Y1 - 2021/4/15

N2 - 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.

AB - 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.

KW - Permutation group

KW - Derangement

KW - Frobenius group

KW - Linear group

U2 - 10.1016/j.jalgebra.2020.12.020

DO - 10.1016/j.jalgebra.2020.12.020

M3 - Article

SN - 0021-8693

VL - 572

SP - 245

EP - 262

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Groups generated by derangements

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Keywords

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