On Groups with Frobenius Elements
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On Groups with Frobenius Elements A. M. POPOV Krasnoyarsk State Technical University, Krasnoyarsk, Russia. e-mail: [email protected] Abstract. It is proved that group G contains an Abelian normal periodic complement to CG (a 2 ) if a is an H -Frobenius element a of order 4 of G. Mathematics Subject Classification (2000): 20E25. Key words: Frobenius group, H -Frobenius element, kernel, complement, involution.
A group G is said to be a Frobenius group with kernel F and complement H if H g is such of G that (1) H ∩ Hx = 1 for every g ∈ G \ H , (2) a set a proper subgroup # x F = x∈G (G \ H ) = G \ x∈G (H \ {1}) is a subgroup, (3) G = F λH [9]. If both G and H meet condition (1) from the definition of a Frobenius group then, according to V. P. Shunkov [9], they form a Frobenius pair (G, H ); Yu. M. Gorchakov [2] suggested another term calling subgroup H peculiar in G. In this paper we’ll use both terms. Finite Frobenius groups have a number of remarkable properties. The orders of their kernel and complement are coprime, i.e. F and H are the Hall subgroups of group G. An arbitrary proper subgroup L of G is contained either in F or in one of the subgroups H x , or it is a Frobenius group with kernel L ∩ F and complement L ∩ H x for some suitable x. According to the Thomson theorem, a kernel F of a finite Frobenius group is nilpotent and its nilpotency length is bounded by a certain function of the minimal prime divisor of |H | (the Higman theorem). It is also known that if H is a locally finite group of regular automorphisms of an Abelian group and 1 (H ) = T is a subroup of H generated by all the elements of prime order, then T is a group of one of the following types: (1) T is a locally cyclic group, (2) T is isomorphic to SL2 (3) or SL2 (5), (3) T = C × S, where C is a group of type 1, S is a group of type 2 and π(C) ∩ π(S) = ∅. Since in the finite case, the structure of Frobenius groups is clear, they can be found as sections in all finite nonnilpotent groups. This makes Frobenius groups a powerful tool for studying groups with various finiteness conditions. The Frobenius groups have played an important role in the study of factorizable, completely factorizable groups and splittable groups with finiteness conditions [10] as well. This research is supported by Krasnoyarsk Science Foundation, grant 11F202C, the Russian
Foundation for Basic Research, grant 03-01-00356.
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The Frobenius groups can be used to answer if there exist any infinite locally finite subgroups in groups [8]. The Frobenius groups and quasifrobenius ones (which are close to the Frobenius groups) are used in characterizations of Chernikov groups [5]. The Frobenius theorem can be fully transferred to the classes of locally finite and binary finite groups (V. M. Busarkin, A. I. Starostin) and to some classes of mixed groups as well (Yu. M. Gorchakov [2]). THEOREM 1 (Yu. M. Gorchakov, [2]). If G = F λH , the subgroup H is locally finite and peculiar in G and the subgroup F is locally nilpotent, then G is a
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