Frobenius Groups Generated by Quadratic Elements |
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Authors: | A. Kh. Zhurtov V. D. Mazurov |
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Affiliation: | (1) Gagarina 205-1, Nalchik, 360016, Russia;(2) Akademika Koptyuga Prospekt, 4, Institute of Mathematics SB RAS, Novosibirsk, 630090, Russia |
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Abstract: | An automorphism of a group X is said to be quadratic if there exist integers and such that for any . If is a Frobenius group then an element is said to be quadratic if induces, by conjugation in the core of , a quadratic automorphism. By definition, a group H acts on a group F freely if for and only with or . It is proved that a Frobenius group generated by two quadratic elements is finite and its core is commutative. In particular, any Frobenius group generated by two elements of order at most 4 is finite. Also we argue that a Frobenius group with finitely generated soluble core is finite. The results mentioned are used to show that a group acting freely on an Abelian group is finite if it is generated by elements of order 3, and the order of a product of every two elements of order 3 in is finite. |
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Keywords: | Frobenius group quadratic automorphism quadratic element |
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