Permutable Subgroups Research Articles

On an application of the lattice of $$\sigma$$-permutable subgroups of a finite group

On an application of the lattice of $$\sigma$$-permutable subgroups of a finite group

Uncountable groups in which permutability is a transitive relation

Abstract Let 𝐺 be a group. A subgroup 𝐻 of 𝐺 is called permutable if H ⁢ K = K ⁢ H HK=KH for all subgroups 𝐾 of 𝐺. Permutability is not in general a transitive relation, and 𝐺 is said to be a PT \mathrm{PT} -group (or to have the PT \mathrm{PT} -property) if, whenever 𝐾 is a permutable subgroup of 𝐺 and 𝐻 is a permutable subgroup of 𝐾, 𝐻 is permutable in 𝐺. The aim of this paper is to investigate the behaviour of uncountable soluble groups of cardinality ℵ whose proper subgroups of cardinality ℵ have the PT \mathrm{PT} -property. We prove that such a group is a PT \mathrm{PT} -group and so are all its subgroups.

Finite groups with some σ-primary subgroups weakly m-ℋ-permutable

Let σ = { σ i | i ∈ I } be some partition of the set of all primes P , G a finite group and σ ( G ) = { σ i | σ i ∩ π ( G ) ≠ ∅ } . A set H of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of H is a Hall σi -subgroup of G for some i and H contains exactly one Hall σi -subgroup of G for every σ i ∈ σ ( G ) . Let H be a complete Hall σ-set of G. A subgroup H of G is said to be H -permutable if HA = AH for every member A ∈ H ; m- H -permutable if H = 〈 A , B 〉 for some modular subgroup A and H -permutable subgroup B of G; weakly m- H -permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ S ≤ H for some m- H -permutable subgroup S of G. In this paper, we study the structure of finite groups G by assuming that some σ-primary subgroups are weakly m- H -permutable in G. Some recent results are generalized and unified.

A generalization of an Agrawal theorem on soluble PST-groups

Let σ = { σ i ⏧ i ∈ I } be some partition of the set of all prime numbers and G a finite group. A subgroup A of G is said to be σ -permutable in G if A permutes with all Hall σi -subgroups H, that is, AH = HA for all i ∈ I . The group G is said to be a P σ T -group if σ -permutability is a transitive relation in G, that is, if K is a σ -permutable subgroup of H and H is a σ -permutable subgroup of G, then K is a σ -permutable subgroup of G. In the case when σ = σ 1 = { { 2 } , { 3 } , { 5 } , … } , a P σ T -group is called a PST-group. We study the structure of finite soluble groups G in which every subnormal subgroup is σ-permutable and G N σ ∩ H ≤ Z ∞ ( H ) for every Hall σi -subgroup H of G and all i. In particular, we obtain a new characterization of soluble P σ T -groups which is a generalization of an Agrawal theorem on soluble PST-groups.

Groups whose proper subgroups of infinite rank have a permutability transitive relation

Abstract Let 𝐺 be a group. A subgroup 𝐻 of 𝐺 is called permutable if H ⁢ X = X ⁢ H HX=XH for all subgroups 𝑋 of 𝐺. Permutability is not in general a transitive relation, and 𝐺 is called a PT \mathrm{PT} -group if, whenever 𝐾 is a permutable subgroup of 𝐺 and 𝐻 is a permutable subgroup of 𝐾, we always have that 𝐻 is permutable in 𝐺. The property PT \mathrm{PT} is not inherited by subgroups, and 𝐺 is called a PT ̄ \overline{\mathrm{PT}} -group if all its subgroups have the PT \mathrm{PT} -property. We prove that if 𝐺 is a soluble group of infinite rank whose proper subgroups of infinite rank have the PT \mathrm{PT} -property, then 𝐺 is a PT ̄ \overline{\mathrm{PT}} -group.

Periodic linear groups factorized by mutually permutable subgroups

The 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.

Periodic linear groups in which permutability is a transitive relation

A 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 mathfrak {X}_p. 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 mathfrak {X}_p 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 mathfrak {X}_p-groups in terms of pronormality (see Theorem 4.11) that will allow us to show that, on some occasions, the property mathfrak {X}_p is inherited by subgroups.

Finite Groups with Hereditarily $$G$$-Permutable Minimal Subgroups

Finite Groups with Hereditarily $$G$$-Permutable Minimal Subgroups

On weakly $s^{\ast}$-permutable subgroups of finite groups

On weakly $s^{\ast}$-permutable subgroups of finite groups

On weakly s p -permutable subgroups of finite groups

Let G be a finite group, H a subgroup of G and p a prime number. We say that H is weakly s p -permutable in G if G has a subnormal subgroup K such that G = HK, and is a -number, where HsG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. In this paper, we investigate the structure of a group G under the assumption that certain subgroups of G are weakly s p -permutable in G. Some recent results are extended and generalized. Communicated by Alexander Olshanskii

The Addition Theorem for two-step nilpotent torsion groups

Abstract The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved in [D. Dikranjan, B. Goldsmith, L. Salce and P. Zanardo, Algebraic entropy for abelian groups, Trans. Amer. Math. Soc. 361 (2009), 7, 3401–3434]. Later, this result was extended to all abelian groups [D. Dikranjan and A. Giordano Bruno, Entropy on abelian groups, Adv. Math. 298 (2016), 612–653] and, recently, to all torsion finitely quasihamiltonian groups [A. Giordano Bruno and F. Salizzoni, Additivity of the algebraic entropy for locally finite groups with permutable finite subgroups, J. Group Theory 23 (2020), 5, 831–846]. In contrast, when it comes to metabelian groups, the additivity of the algebraic entropy fails [A. Giordano Bruno and P. Spiga, Some properties of the growth and of the algebraic entropy of group endomorphisms, J. Group Theory 20 (2017), 4, 763–774]. Continuing the research within the class of locally finite groups, we prove that the Addition Theorem holds for two-step nilpotent torsion groups.

A generalization of some results of the theory of permutable subgroups

A generalization of some results of the theory of permutable subgroups

On Groups Factorized by Mutually Permutable Subgroups

The aim of the paper is to provide a large extension of the recent results of de Giovanni and Ialenti (Commun Algebra 44:118–124, 2016), strengthening at the same time their conclusions. Our second main theorem is actually a complete generalization of a result obtained in the finite case by Beidleman and Heineken (Arch Math (Basel) 85:18–30, 2005) to periodic linear groups.

On the w-supersolubility of a finite group factorized by mutually permutable subgroups

On the w-supersolubility of a finite group factorized by mutually permutable subgroups

On weakly σ-semipermutable subgroups of finite groups

Let [Formula: see text] be some partition of the set of all primes [Formula: see text], G be a finite group and [Formula: see text]. A set [Formula: see text] of subgroups of [Formula: see text] is said to be a complete Hall [Formula: see text]-set of G if every non-identity member of [Formula: see text] is a Hall [Formula: see text]-subgroup of G for some i and [Formula: see text] contains exactly one Hall [Formula: see text]-subgroup of G for every [Formula: see text]. Let [Formula: see text] be a complete Hall [Formula: see text]-set of G. A subgroup A of G is said to be [Formula: see text]-semipermutable with respect to [Formula: see text] if [Formula: see text] for all [Formula: see text] and all [Formula: see text] such that [Formula: see text]; [Formula: see text]-semipermutable in G if A is [Formula: see text]-semipermutable in G with respect to some complete Hall [Formula: see text]-set of G. We say that a subgroup H of G is weakly [Formula: see text]-semipermutable in G if there exists a [Formula: see text]-permutable subgroup [Formula: see text] of [Formula: see text] such that HT is [Formula: see text]-permutable in G and [Formula: see text], where [Formula: see text] is the subgroup of H generated by all those subgroups of H which are [Formula: see text]-semipermutable in G. In this paper, we study the structure of G under the condition that some subgroups of G are weakly [Formula: see text]-semipermutable in G.

The supersolvable residual of a finite group factorized by pairwise permutable seminormal subgroups

The supersolvable residual of a finite group factorized by pairwise permutable seminormal subgroups

On -permutable subgroups in finite groups

UDC 512.542 Let be some partition of the set of all primes and let be a nonempty subset of the set A set of subgroups of a finite group is said to be a \emph{complete Hall -set} of if every member of is a Hall -subgroup of for some and contains exactly one Hall -subgroup of for every such that A subgroup of is called (i) {-<em>permutable</em>} if for and ; (ii) {-<em>permutable</em> in } if is -permutable for some complete Hall -set of We study the influence of -permutable subgroups on the structure of In particular, we prove that if and where and are -permutable -separable (respectively, -closed) subgroups of then is also -separable (respectively, -closed). Some known results are generalized.

The Supersolvable Residual of a Finite Group Factorized by Pairwise Permutable Seminormal Subgroups

The Supersolvable Residual of a Finite Group Factorized by Pairwise Permutable Seminormal Subgroups

Permutable Subgroups in GLn(D) and Applications to Locally Finite Group Algebras

In this paper we study the existence of free nonabelian subgroups in noncentral permutable subgroups of general skew linear groups and locally finite group algebras.

Finite groups with generalized subnormal and generalized permutable subgroups

Let [Formula: see text] be a finite group and [Formula: see text] a subgroup of [Formula: see text]. We say that [Formula: see text] is strongly[Formula: see text]-subnormal in [Formula: see text] if either [Formula: see text] is normal in [Formula: see text] or [Formula: see text] and every chief factor of [Formula: see text] between [Formula: see text] and [Formula: see text] is cyclic; strongly[Formula: see text]-permutable in [Formula: see text] if [Formula: see text] permutes with all Hall [Formula: see text]’-subgroups and with all Sylow [Formula: see text] -subgroups of [Formula: see text] for all [Formula: see text]; partially[Formula: see text]-subnormal in [Formula: see text] if [Formula: see text], where [Formula: see text] is a strongly [Formula: see text]-subnormal subgroup and [Formula: see text] is a [Formula: see text]-subnormal subgroup of [Formula: see text]. In this paper, we study finite groups whose 3-maximal or 2-maximal subgroups are partially [Formula: see text]-subnormal (strongly [Formula: see text]-permutable).