### Abstract

We study the cycle structure of those permutations of the finite field F_{qn} of the form L(x) = ∑ i=0 n-1a_{i}x^{qi} where each a_{i} ε{lunate} F_{q}. For such a permutation, the problem of finding its cycle decomposition of F_{qn} can be reduced to finding its cycle decomposition on certain T-invariant subspaces of F_{qn}, where T is the operator defined by T : x → x^{q}. If L_{1}(x) and L_{2}(x)M are in the above form, we say that L_{1}(x) and L_{2}(x) are equivalent if L_{1}(x) and L_{2}(x) induce the same cycle decomposition of F_{qn}, and we say they are strongly equivalent if they induce the same cycle decomposition in every T-invariant subspace of F_{qn}. We show that these notions are not the same, and we give characterizing theorems for each.

Original language | English (US) |
---|---|

Pages (from-to) | 63-82 |

Number of pages | 20 |

Journal | Linear Algebra and Its Applications |

Volume | 108 |

Issue number | C |

DOIs | |

State | Published - Sep 1988 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Cite this

*Linear Algebra and Its Applications*,

*108*(C), 63-82. https://doi.org/10.1016/0024-3795(88)90179-6

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*Linear Algebra and Its Applications*, vol. 108, no. C, pp. 63-82. https://doi.org/10.1016/0024-3795(88)90179-6

**Cycles of linear permutations over a finite field.** / Mullen, Gary L.; Vaughan, Theresa P.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Cycles of linear permutations over a finite field

AU - Mullen, Gary L.

AU - Vaughan, Theresa P.

PY - 1988/9

Y1 - 1988/9

N2 - We study the cycle structure of those permutations of the finite field Fqn of the form L(x) = ∑ i=0 n-1aixqi where each ai ε{lunate} Fq. For such a permutation, the problem of finding its cycle decomposition of Fqn can be reduced to finding its cycle decomposition on certain T-invariant subspaces of Fqn, where T is the operator defined by T : x → xq. If L1(x) and L2(x)M are in the above form, we say that L1(x) and L2(x) are equivalent if L1(x) and L2(x) induce the same cycle decomposition of Fqn, and we say they are strongly equivalent if they induce the same cycle decomposition in every T-invariant subspace of Fqn. We show that these notions are not the same, and we give characterizing theorems for each.

AB - We study the cycle structure of those permutations of the finite field Fqn of the form L(x) = ∑ i=0 n-1aixqi where each ai ε{lunate} Fq. For such a permutation, the problem of finding its cycle decomposition of Fqn can be reduced to finding its cycle decomposition on certain T-invariant subspaces of Fqn, where T is the operator defined by T : x → xq. If L1(x) and L2(x)M are in the above form, we say that L1(x) and L2(x) are equivalent if L1(x) and L2(x) induce the same cycle decomposition of Fqn, and we say they are strongly equivalent if they induce the same cycle decomposition in every T-invariant subspace of Fqn. We show that these notions are not the same, and we give characterizing theorems for each.

UR - http://www.scopus.com/inward/record.url?scp=50849145572&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=50849145572&partnerID=8YFLogxK

U2 - 10.1016/0024-3795(88)90179-6

DO - 10.1016/0024-3795(88)90179-6

M3 - Article

AN - SCOPUS:50849145572

VL - 108

SP - 63

EP - 82

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - C

ER -