## 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

Zbl 0486.20016
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.