SUBGROUPS OF FINITE GROUPS WITH A STRONG COVER-AVOIDANCE PROPERTY

AbstractA subgroup A of a group G has the strong cover-avoidance property in G, or A is a strong CAP-subgroup of G, if A either covers or avoids every chief factor of every subgroup of G containing A. The main aim of the present paper is to analyse the impact of the strong cover and avoidance property of the members of some relevant families of subgroups on the structure of a group.

A Characterization of the Class of Finite Groups with Nilpotent Derived Subgroup

The class of all finite groups with nilpotent commutator subgroup is characterized as the largest subgroup-closed saturated formation 𝔉 for which the 𝔉-residual of a group generated by two 𝔉-subnormal subgroups is the subgroup generated by their 𝔉–residuals.

A question in the theory of saturated formations of finite soluble groups

This paper examines the following question. If\(\mathcal{H}\) and\(\mathcal{F}\) are saturated formations then\(\mathcal{H}_\mathcal{F} \) is defined to be the class of all soluble groups whose\(\mathcal{H} - normalizers\) belong to\(\mathcal{F}\). In general\(\mathcal{H}_\mathcal{F} \) is a formation, but need not be a saturated formation. Here the smallest saturated formation containing\(\mathcal{H}_\mathcal{F} \) is studied.

On Join Properties of Hall π-Subgroups of Finite π-Soluble Groups

All groups considered in the sequel are finite. K. Doerk and T. Hawkes, in Section I.4 of their recent comprehensive w x volume on finite soluble groups 1 , include background material and a proof of the following result: Let S be a Hall system of a soluble group G and let U and V be subgroups into which S reduces. Then S reduces into U l V, and if , in addition, U permutes with V, then S reduces into UV. It is clear that the second part of the above result holds equally well with a single Hall subgroup in place of a Hall system; in other words, if a Hall p-subgroup of G contains Hall p-subgroups of U and V and U permutes with V, then it also contains a Hall p-subgroup of UV.

On the focal subgroup of a saturated fusion system

Abstract The influence of the cyclic subgroups of order p or 4 of the focal subgroup of a saturated fusion system F over a p -group S is investigated in this paper. Some criteria for normality of S in F as well as necessary and sufficient conditions for nilpotency of F are given. The resistance of a p -group in which every cyclic subgroup of order p or 4 is normal, and earlier results about p -nilpotence of finite groups and nilpotency of saturated fusion systems are consequences of our study.

On partial CAP-subgroups of finite groups

Abstract Given a chief factor H / K of a finite group G, we say that a subgroup A of G avoids H / K if H ∩ A = K ∩ A ; if H A = K A , then we say that A covers H / K . If A either covers or avoids the chief factors of some given chief series of G, we say that A is a partial CAP-subgroup of G. Assume that G has a Sylow p-subgroup of order exceeding p k . If every subgroup of order p k , where k ≥ 1 , and every subgroup of order 4 (when p k = 2 and the Sylow 2-subgroups are non-abelian) are partial CAP-subgroups of G, then G is p-soluble of p-length at most 1.

On second maximal subgroups of Sylow subgroups of finite groups

Abstract Finite groups in which the second maximal subgroups of the Sylow p -subgroups, p a fixed prime, cover or avoid the chief factors of some of its chief series are completely classified.

On the p-length of some finite p-soluble groups

The main aim of this paper is to give structural information of a finite group of minimal order belonging to a subgroup-closed class of finite groups and whose $p$-length is greater than $1$, $p$ a prime number. Alternative proofs and improvements of recent results about the influence of minimal $p$-subgroups on the $p$-nilpotence and $p$-length of a finite group arise as consequences of our study

Triple Factorizations and Supersolubility of Finite Groups

AbstractIn this paper we analyse the structure of a finite group of minimal order among the finite non-supersoluble groups possessing a triple factorization by supersoluble subgroups of pairwise relatively prime indices. As an application we obtain some sufficient conditions for a triple factorized group by supersoluble subgroups of pairwise relatively prime indices to be supersoluble. Many results appear as consequences of our analysis.

On saturated formations of finite groups with a restricted cover and avoidance property

On Formations of Finite Groups with the Wielandt Property for Residuals

Abstract Given two subgroups U, V of a finite group which are subnormal subgroups of their join 〈U, V〉 and a formation F , in general it is not true that 〈U, V〉 F  = 〈U F , V F 〉. A formation is said to have the Wielandt property if this equality holds universally. A formation with the Wielandt property must be a Fitting class. Wielandt proved that the most usual Fitting formations (e.g., nilpotent groups and π-groups) have the Wielandt property. At present, neither a general satisfactory result on the universal validity of the Wielandt property nor a counterexample is known. In this paper a criterion for a Fitting formation to have the Wielandt property is given. As an application, it is p…

A note on finite groups generated by their subnormal subgroups

AbstractFollowing the theory of operators created by Wielandt, we ask for what kind of formations $\mathfrak{F}$ and for what kind of subnormal subgroups $U$ and $V$ of a finite group $G$ we have that the $\mathfrak{F}$-residual of the subgroup generated by two subnormal subgroups of a group is the subgroup generated by the $\mathfrak{F}$-residuals of the subgroups.In this paper we provide an answer whenever $U$ is quasinilpotent and $\mathfrak{F}$ is either a Fitting formation or a saturated formation closed for quasinilpotent subnormal subgroups.AMS 2000 Mathematics subject classification: Primary 20F17; 20D35