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On Frobenius pairs of the form (F\(\leftthreetimes V,V)\). (English. Russian original) Zbl 0722.20027
Math. USSR, Sb. 68, No. 1, 165-179 (1990); translation from Mat. Sb. 180, No. 10, 1311-1324 (1989).
The author is well-known for his results giving criteria for a group to be finite or locally finite. This paper contains the following theorem, which was used in his preprint [Groups with involutions, part II (Prepr. No.5, VC SO AN SSSR, Krasnoyarsk (1986)] for the characterization of finite groups of even order. Theorem. Let G be a group, H a proper subgroup, \(V=gr(\{x,y\}^ H)\), where x and y are some nontrivial elements of finite order in H, at least one of which has even order and at least one of which is not an element of order 2, which satisfy the following additional conditions: a) \(V\cap V^ g=1\) (g\(\in G\setminus H)\). b) For almost all (i.e., all but possibly a finite number) elements of the form \(x^ g\) (g\(\in G\setminus H)\) the subgroups \(gr(x,x^ g)\) are finite. c) If for some \(s\in G\setminus H\) the subgroup \(gr(x,y^ s)\) is finite, then \[ H\cap gr(x,y^ s),H\cap gr(x^{s^{-1}},y)\leq V. \] Then \(G=F\leftthreetimes C_ G(i)\), where i is an element of order 2 in V, \(H=C_ G(i)\), F is a periodic subgroup, and (F\(\leftthreetimes V,V)\) is a Frobenius pair.
The author remarks that in the case of finite groups this result and its corollary are natural complements to Glauberman’s \(Z^*\)-theorem. Corollary. Let G be a group, H a proper subgroup, and \(V=gr(\{x\}^ H)\), where x is some element of even order \(>2\) in H, satisfying the following additional conditions: a) \(V\cap V^ g=1\) (g\(\in G\setminus H)\). b) For almost all (i.e. all but, possibly, a finite number) elements of the form \(x^ g\) (g\(\in G\setminus H)\) the subgroups \(gr(x,x^ g)\) are finite. c) If for some \(s\in G\setminus H\) the subgroup \(gr(x,x^ s)\) is finite, then \(H\cap gr(x,x^ s)\leq V\). Then \(G=F\leftthreetimes C_ G(i)\), where i is an element of order 2 in V, \(H=C_ G(i)\), F is a periodic abelian subgroup, and (F\(\leftthreetimes V,V)\) is a Frobenius pair. It is also shown by means of examples that conditions a), b), and c) are not dependent on each other and that each of them is an essential constraint in the theorem.
MSC:
20F50 Periodic groups; locally finite groups
20E25 Local properties of groups
20F05 Generators, relations, and presentations of groups
20E34 General structure theorems for groups
20E07 Subgroup theorems; subgroup growth
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