On conditional permutability and factorized groups

Abstract

Two subgroups A and B of a group G are said to be totally completely conditionally permutable (tcc-permutable) if X permutes with Y g for some g ∈ X, Y , for all X ≤ A and all Y ≤ B. In this paper we study finite products of tcc-permutable subgroups, focussing mainly on structural properties of such products. As an application, new achievements in the context of formation theory are obtained.

Conditional permutability of subgroups and certain classes of groups

“…As mentioned in the Introduction (see also Lemmas 4,5), the structure of tcc-permutable products which are monolithic primitive groups has been described in [4,Lemma 4,Corollary 5]. As an application of our results on the cover-avoidance property, the structure of a tcc-permutable product of two factors, which is a non-monolithic primitive group, is also clarified.…”

“…Though it should be mentioned that this fact is also a consequence of a stronger result [26,Theorem 3.8] involving c-permutability, which shows in particular that a group is supersoluble if and only if every maximal subgroup is c-permutable in the group. The second part of the next Corollary 3 is a consequence of a significant property of tcc-permutable products, obtained in [4,Theorem 3], which states that the nilpotent residuals of the corresponding factors are normal subgroups in the whole group. With the previous notation, since A = A(A ∩ B) and B = B(A ∩ B) are tcc-permutable products of the subgroups A and A ∩ B, and B and A ∩ B, respectively, we can deduce that G = AB normalizes (A ∩ B) N .…”

“…The present paper is mainly concerned with tcc-permutability, and we are interested in a better understanding of the inner structure of such products of subgroups, particularly in their normal structure. In this direction, we take further the study carried out in [4,6], and are inspired by the investigation of Beidleman and Heineken in [17] on mutually permutable products. As in the case of mutually permutable products (see [17]), we prove in Section 2 that the factors of a tcc-permutable product have the cover-avoidance property in the product (Corollary 1), i.e.…”

“…The results in Section 2 are then applied in Section 3 to find new classes of groups which are closed with respect to tcc-permutable products. A previous research within the framework of formation theory has been carried out in [4,6]. Now, we take further previous developments on totally and mutually permutable products, and search for the interaction between the ascending series of classes of T-, PT-, PST-, SC-, and SM-groups and tcc-permutability.…”

“…We point out that totally permutable products are tcc-permutable, and so, in many cases, the research on the last kind of products develops as generalization of previous studies on the first ones. It is remarkable how many results on totally permutable products remain true in the more general setting in spite of the failure of significant structural properties (see [2,4,6] and Remark 1). As showed in this paper, even more surprising is the revealed analogy of certain results between mutually permutable products and tccpermutable products.…”