Siberian Mathematical Journal, Vol. 38, No. 6, 1997 ON FROBENIUS GROUPS t) V. V. Bludov UDC 512.54 In 1978 V. P. Shunkov raised the following problems in [1, problem 6.53]: ``What can we say about the kernel and complement of a Frobenius group? In particular, which groups can serve as kernel and complement?" Furthermore, in [1, problem 13.54b] A. I. Sozutov asked: "Is it true that every group embeds in the kernel of some Frobenius group?" In the article [2], A. I. Sozutov described weakly conjugately biprimitively finite groups that may serve as complements in some Frobenius group with abelian kernel. This gives a partial answer to V. P. Shtmkov's problem. In the present article, we prove the following: (1) every group embeds in the kernel of a suitable Frobenitm group; (2) under such embedding, the complement can be chosen to be any nontrivial right-ordered group (Theorem 1). The first result gives a positive answer to A. I. Sozutov's question. As far as the second result is concerned, it is likely that the following stronger assertion is valid which we propose as a conjecture: every group embeds in the kernel o[ some b'~obenius group; moreover, the complement can be taken to be an arbitrary nontrivial torsion-free group. DEFINITION 1. The semidireet product G = FAH of nontrivial groups F and H is called the Frobenius group with kernel F and complement H if the following conditions are satisfied: H N HJ = e forall E G \ H, (1) F \ {e} = G \ U Hg" (2) geG It is well known that condition (1) is equivalent to regularity of the action of the subgroup H on the kernel F by conjugation: fh # f for any f e F \ {e}, h e H \ {e}, which is in turn equivalent to iajectivity of the mapping ah : F ~ F, ah(f) -" If, h], for all h E H \ {e} (see [3]). Factually, condition (1) of the semidirect product is equivalent to injectivity, and condition (2) to surjectivity, of the mapping a~ for all h e H \ {e} (see [4, 5]). Therefore, Definition 1 acquires the following equivalent form: the semidirect product G = FAH of nontrivial groups F and H is the Frobenius group with kernd F and complemeat H ff and only if the mapping a~ : F ~ F, a~(f) = [f, h], is bijective for al/h E H \ {e}. DEFINITION 2. A right-ordered group is an algebraic system (G,-, <) satisfying the conditions: (R1) (C,.) is a group; (R2) (G, <) is a linearly ordered set; (R3) a < b =~ ac < be for all a, b, c e G. It is immediate from the definition that a right-ordered group is torsion-free. The converse fails in general (see [6]). In [7] (also see [6]) the following characterization of right-ordered groups was given: every right-ordered group is isomorphic to some subgroup o[ the group of all order automorphisms of a suitable linearly ordered set (or a linearly ordered free abelian group). Theorem 1. Every group embeds in the kernel of some Frobenius group. Moreover, the comple- ment may be any nontrivial right-ordered group. PROOF. Let H be a nontrivial right-ordered group and let IV(H) = {W C HI W = o or (W, <) is a totally ordered set}. t) The research was supported by the Russian Foundation for Basic Research (Grant 96-01-00358). Irkutsk. Translated from Sibirskii Matemalichesk~ Zhurnal, Vol. 38, No. 6, pp. 1219-1221, November-December, 1997. Original article submitted 3une 6, 1996. 1054 0037-4466/97/3806--1054 $18.00 (~ 1997 Plenum PubRshin 8 Corporation