Primitive permutation groups and derangements of prime power order. (English) Zbl 1345.20002
Let \(G\) be a transitive permutation group on a finite set \(\Omega\) of size at least 2. An element \(x\in G\) is a derangement if it acts fixed-point-freely on \(\Omega\). By a theorem of B. Fein et al. [J. Reine Angew. Math. 328, 39-57 (1981; Zbl 0457.13004)], \(G\) contains a derangement of prime power order. In this paper, the authors study the finite primitive permutation groups with the extremal property that every derangement has order a power of \(r\) for some fixed prime \(r\). Earlier, I. M. Isaacs et al. [J. Pure Appl. Algebra 207, No. 3, 717-724 (2006; Zbl 1111.20002)] described the finite transitive permutation groups in which every derangement has order 2.
First, the authors show that considered groups are either almost simple or affine, and determine all the almost simple groups with this property. Then, they prove that an affine group \(G\) has this property if and only if every two-point stabilizer is an \(r\)-group. Here the structure of \(G\) has been extensively studied in works of R. Guralnick and R. Wiegand [Trans. Am. Math. Soc. 331, No. 2, 563-584 (1992; Zbl 0753.12003)] and P. Fleischmann et al. [J. Algebra 188, No. 2, 547-579 (1997; Zbl 0896.20007)].
First, the authors show that considered groups are either almost simple or affine, and determine all the almost simple groups with this property. Then, they prove that an affine group \(G\) has this property if and only if every two-point stabilizer is an \(r\)-group. Here the structure of \(G\) has been extensively studied in works of R. Guralnick and R. Wiegand [Trans. Am. Math. Soc. 331, No. 2, 563-584 (1992; Zbl 0753.12003)] and P. Fleischmann et al. [J. Algebra 188, No. 2, 547-579 (1997; Zbl 0896.20007)].
Reviewer: Anatoli Kondrat’ev (Ekaterinburg)
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