Abstract
Let G be a finite group and m a natural number. The twisting function with m variables on G is defined by \({\tau_m(x_1,\ldots, x_m) :=(x_1^{x_2}, \ldots, x_{m-1}^{x_m}, x_m^{x_1})}\) and has been introduced and studied by the third author in [5]. In the current paper, we extend and sharpen results from Kaplan (see [5]) on solvability and nilpotency (see Theorems 1.2 and 1.1). Furthermore, for groups G such that \({\tau_2}\) is a permutation on the cartesian product G 2, we investigate the order of this permutation and its connection to properties of G.
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References
Baer R.: Group Elements of Prime Power Index. Trans. Amer. Math. Soc. 75, 20–47 (1953)
J.D. Dixon and B. Mortimer, Permutation groups, Springer-Verlag 1996.
D. Gorenstein, Finite Groups, Chelsea 1980
B. Huppert, Endliche Gruppen I, Springer-Verlag 1967
G. Kaplan, Nilpotency, Solvability and the Twisting Function of Finite Groups, Comm. Alg. 39 (2011), 1722–1729.
M. Suzuki, Group Theory I, Springer-Verlag 1982.
Thompson J.G.: Nonsolvable finite groups all of whose local subgroups are solvable. Bull. Amer. Math. Soc. 74, 383–437 (1968)
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Dedicated to O.H. Kegel on the occasion of his 80th birthday
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Heineken, H., Herfort, W. & Kaplan, G. Nilpotency, solvability and the twisting function of finite groups II. Arch. Math. 102, 501–512 (2014). https://doi.org/10.1007/s00013-014-0656-y
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DOI: https://doi.org/10.1007/s00013-014-0656-y