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Generalized Frobenius groups. (English) Zbl 0533.20012

In this paper the authors develop some results related to a generalization of Frobenius groups due to the reviewer [A. R. Camina, Isr. J. Math. 31, 153-160 (1978)]. A finite group G was said to satisfy (F2) if G possesses a proper non-trivial normal subgroup K such that if \(x\in G/K\) and z is in K, then \([x,y]=z\) for some y in G. The reviewer showed that if G satisfies (F2) then either G is a Frobenius group or at least one of G/K or K is a p-group for some prime p. The second author [I. D. Macdonald, Isr. J. Math. 40, 350-364 (1981; Zbl 0486.20016)] has studied the structure of such groups when G is a p- group.
The authors study the situation when G satisfies (F2) and is neither a Frobenius group nor a p-group. In such a situation either G/K or K is a p-group for some prime p but not both. When this occurs they say that (G,K) is an F2(p) pair. In section 2 they give some new examples of such pairs. The main results are proved in sections 4 and 5. Theorem 4.1. Let (G,K) have F2(p) with K a p-group for some prime p. Let P be a Sylow p- subgroup of G. Assume one of the following conditions holds: (i) G is p- soluble and P/K is Abelian. (ii) G is a p-soluble and P has nilpotency class at most 2. (iii) K is cyclic. Then P is normal in G and (P,K) has (F2). G is a semidirect product \(G=HP\), and HK is a Frobenius group with kernel K.
Theorem 5.1. Let (G,K) have F2(p) with G/K a p-group, for some prime p. If a Sylow p-subgroup of G has class at most 2, then G is Frobenius, the Frobenius kernel has index 2 in K and the Frobenius complement is the quaternion group of order 8. In particular \(p=2\) and \(| G/K| =4\). - These two results go a lot further than previous work but leave a lot of interesting unanswered questions.
Reviewer: A.R.Camina

MSC:

20D25 Special subgroups (Frattini, Fitting, etc.)
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D40 Products of subgroups of abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks

Citations:

Zbl 0486.20016
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References:

[1] A. R. Camina,Some conditions which almost characterize Frobenius groups Isr. J. Math.31 (1978), 153–160. · Zbl 0654.20019
[2] D. Gorenstein,Finite Groups, Harper and Row, New York-London, 1968.
[3] I. M. Isaacs,Character Theory of Finite Groups, Academic Press, New York-San Francisco-London, 1976. · Zbl 0337.20005
[4] J. D. Macdonald,Some p-groups of Frobenius and extra-special type, Isr. J. Math.40 (1981), 350–364. · Zbl 0486.20016
[5] D. J. S. Robinson,A Course in the Theory of Groups, Springer-Verlag, New York-Berlin-heidelberg, 1982. · Zbl 0483.20001
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