1,721,021 research outputs found
A note on conjugacy of supplements in soluble periodic linear groups
No full textThe aim of this short note is to prove that if G is a (homomorphic images of a) soluble periodic linear group and N is a locally nilpotent normal subgroup of G such that N and G/N have no isomorphic G-chief factors, then two supplements to N in G are conjugate provided that they have the same intersection with N. This result follows from well-known theorems in the theory of Schunck classes (see [A. Ballester-Bolinches and L. M. Ezquerro, On conjugacy of supplements of normal subgroups of finite groups, Bull. Aust. Math. Soc. 89 (2014), no. 2, 293-299]), and it appeared as the main theorem of [C. Parker and P. Rowley, A note on conjugacy of supplements in finite soluble groups, Bull. Lond. Math. Soc. 42 (2010), no. 3, 417-419]
Paranilpotency in uncountable groups
No full textThe aim of this paper is to provide a contribution to the theory of uncountable groups and to that of paranilpotent groups. Extending the structural results in Franciosi and de Giovanni (Ricerche Mat 40:321–333, 1991) and de Giovanni et al. (Comm Algebra 49:3020–3033, 2021), we prove that locally soluble minimal non-paranilpotent groups, i.e. non-paranilpotent groups whose proper subgroups are paranilpotent, are soluble. It is also shown that the class of paranilpotent groups is countably recognizable and, as an application of these results, that a soluble uncountable group whose proper uncountable subgroups are paranilpotent is itself paranilpotent
Infinite groups with many complemented subgroups
No full textThis paper has two souls. On one side, it is a survey on (infinite) groups in which certain systems of subgroups are complemented (like for instance the abelian subgroups). On another side, it provides generalizations and new, easier proofs of some (un)known results in this area
Periodic linear groups in which permutability is a transitive relation
No full textA PT -group is a group in which the relation of being a permutable subgroup is transitive. The main aim of this paper is to show that a (homomorphic image of a) periodic linear group is a soluble PT-group if and only if each subgroup of a Sylow subgroup is permutable in the corresponding Sylow normalizer (see Theorem 4.7); for a fixed prime p, the latter condition is denoted by Xp . In order to prove our main theorem, we need (i) to characterize (homomorphic images of) periodic linear groups that are PT-groups (see Sect. 2), (ii) to develop a fusion theory for locally finite groups (see Sect. 3), (iii) to carefully study (homomorphic images of) periodic linear groups with the property Xp for a fixed prime p (see for instance Theorem 4.6). As a by-product we obtain (among other results) a characterization of (homomorphic images of) periodic linear Xp -groups in terms of pronormality (see Theorem 4.11) that will allow us to show that, on some occasions, the property Xp is inherited by subgroups
Periodic linear groups factorized by mutually permutable subgroups
No full textThe aim is to investigate the behaviour of (homomorphic images of) periodic linear groups which are factorized by mutually permutable subgroups. Mutually permutable subgroups have been extensively investigated in the finite case by several authors, among which, for our purposes, we only cite J. C. Beidleman and H. Heineken (2005). In a previous paper of ours (see M. Ferrara, M. Trombetti (2022)) we have been able to generalize the first main result of J. C. Beidleman, H. Heineken (2005) to periodic linear groups (showing that the commutator subgroups and the intersection of mutually permutable subgroups are subnormal subgroups of the whole group), and, in this paper, we completely generalize all other main results of J. C. Beidleman, H. Heineken (2005) to (homomorphic images of) periodic linear groups
Countable and Uncountable in Group Theory
Full text linkA prominent, recurring feature of group theory has been the determination of groups (all of) whose subgroups possess some group theoretical property. For infinite groups in a suitable universe a number of different approaches have been used in this regard. For locally finite groups, for example, knowledge of the structure of the finite subgroups is often crucial. On the other hand the concept of "largeness" has also recently played an interesting role. Moving from this, I started to study how subgroups of uncountable cardinality affect an uncountable group. Let X be a group theoretical proper, let G be a group of uncountable cardinality and suppose that all its proper uncountable subgroups satisfy X. Is it true that all (proper) subgroups of G satisfy X? The thesis exploits this question, showing that, under some soluble conditions, the answer is often positive. Finally the thesis deals with countably recognizable properties, which has a strong relation with the previous question
Groups with iterated restrictions on conjugacy classes
No full textLet be a group class (such as the class of all finite groups). Starting from , we can define the class C of all groups G such that, for any g G, the co-centralizer G/CG((g)G) of g in G is an -group; of course, if = , these are the well-known FC-groups. Iterating this request, we define the class C2 of groups whose co-centralizers are C-groups, and so on. We generically refer to these groups as groups with -iterated conjugacy classes. Of course, if is quotient closed, then any group G such that G/ζk(G) , for some k ≥ 0, has -iterated conjugacy classes, and actually these concepts are almost always equivalent in the universe of linear groups. For = , this type of restrictions have recently been investigated, and the aim of this paper is to study the general theory of groups with -iterated conjugacy classes, paying particular attention to the case in which is the class C of Cernikov groups: we extend (and improve) results concerning groups with -iterated conjugacy classes. The main focus is on Sylow theory, serial subgroups and groups with many proper subgroups having C-iterated conjugacy classes
Locally finite simple groups whose non-Abelian subgroups are pronormal
No full textA subgroup X of a group G is said to be pronormal if the subgroups X and (Formula presented.) are conjugate in (Formula presented.) for all (Formula presented.). Moreover, in analogy with metahamiltonian groups (i.e. groups in which every non-abelian subgroup is normal), a group in which every non-abelian subgroup is pronormal is called prohamiltonian. In this article we will determine those finite simple groups which are prohamiltonian. It will easily follow that the only prohamiltonian locally finite simple groups are the finite ones
A local study of group classesi
No full textIt was established in [15] that the class of groups with a finite commutator subgroup can be locally described by locally graded groups having a bound on the length of particular chains of non-normal subgroups. This approximation was later extended to groups having a finite normal subgroup whose factor group has no non-permutable subgroups (see [18]). The aim of this paper is to show that these approximating group classes behave better than the classes they approximate, and can be used to derive new results on these
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