Arithmetic functions and fixed points of powers of permutations. (English) Zbl 1534.11118
Author’s abstract: Let \(\sigma\) be a permutation of a nonempty finite or countably infinite set \(X\) and let \(F_X(\sigma^k)\) count the number of fixed points of the \(k\)th power of \(\sigma\). This paper explains how the arithmetic function \(k\mapsto (F_X(\sigma^k))_{k=1}^{\infty}\) determines the conjugacy class of the permutation \(\sigma\), constructs an algorithm to compute the conjugacy class from the fixed point counting function \(F_X(\sigma^k)\), and describes the arithmetic functions that are fixed point counting functions of permutations.
Reviewer: László Tóth (Pécs)
MSC:
11N56 | Rate of growth of arithmetic functions |
20B05 | General theory for finite permutation groups |
20B07 | General theory for infinite permutation groups |
20B10 | Characterization theorems for permutation groups |
20F69 | Asymptotic properties of groups |
References:
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[2] | Diaconis, P.; Fulman, J.; Guralnick, R., On fixed points of permutations, J. Algebraic Combin., 28, 189-218 (2008) · Zbl 1192.20001 · doi:10.1007/s10801-008-0135-2 |
[3] | Ford, K.: Cycle type of random permutations: a toolkit. Discrete Anal. 2022, Paper No. 9, 36 pp. (2022) |
[4] | Nathanson, MB, Elementary Methods in Number Theory (2000), New York: Springer, New York · Zbl 0953.11002 |
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