Groups in which each subnormal subgroup is commensurable with some normal subgroup

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ulderico Dardano ◽  
Fausto De Mari
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Soluble Group ◽  
Study Groups ◽  
Subnormal Subgroup

Abstract We study groups in which each subnormal subgroup is commensurable with a normal subgroup. Recall that two subgroups 𝐻 and 𝐾 are termed commensurable if H ∩ K H\cap K has finite index in both 𝐻 and 𝐾. Among other results, we show that if a (sub)soluble group 𝐺 has the above property, then 𝐺 is finite-by-metabelian, i.e., G ′′ G^{\prime\prime} is finite.

scholarly journals Cyclically separated groups

1982 ◽  
Vol 26 (3) ◽  
pp. 355-384 ◽  
Author(s):  
Brian Hartley ◽  
John C. Lennox ◽  
Akbar H. Rhemtulla
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Finite Rank ◽  
Soluble Group ◽  
Residually Finite Groups ◽  
Interesting Class ◽  
Isolated Subgroup ◽  
Free Section ◽  
Torsion Free ◽  
Free Soluble Group

We call a group G cyclically separated if for any given cyclic subgroup B in G and subgroup A of finite index in B, there exists a normal subgroup N of G of finite index such that N ∩ B = A. This is equivalent to saying that for each element x ∈ G and integer n ≥ 1 dividing the order o(x) of x, there exists a normal subgroup N of G of finite index such that Nx has order n in G/N. As usual, if x has infinite order then all integers n ≥ 1 are considered to divide o(x). Cyclically separated groups, which are termed “potent groups” by some authors, form a natural subclass of residually finite groups and finite cyclically separated groups also form an interesting class whose structure we are able to describe reasonably well. Construction of finite soluble cyclically separated groups is given explicitly. In the discussion of infinite soluble cyclically separated groups we meet the interesting class of Fitting isolated groups, which is considered in some detail. A soluble group G of finite rank is Fitting isolated if, whenever H = K/L (L ⊲ K ≤ G) is a torsion-free section of G and F(H) is the Fitting subgroup of H then H/F(H) is torsion-free abelian. Every torsion-free soluble group of finite rank contains a Fitting isolated subgroup of finite index.

scholarly journals Multidimensional Multiplicative Combinatorial Properties of Dynamical Syndetic Sets

2021 ◽  
Author(s):  
Jiahao Qiu ◽  
Jianjie Zhao
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Minimal System ◽  
Combinatorial Properties ◽  
Return Times ◽  
Residual Set ◽  
Closed Sets ◽  
Open Set ◽  
Geometric Progressions ◽  
Set Of Points

AbstractIn this paper, it is shown that for a minimal system (X, G), if H is a normal subgroup of G with finite index n, then X can be decomposed into n components of closed sets such that each component is minimal under H-action. Meanwhile, we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension, extending a previous result by Glasscock, Koutsogiannis and Richter.

scholarly journals Averaging formula for Nielsen coincidence numbers

2007 ◽  
Vol 186 ◽  
pp. 69-93 ◽  
Author(s):  
Seung Won Kim ◽  
Jong Bum Lee
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Holonomy Group ◽  
Smooth Manifolds

AbstractIn this paper we study the averaging formula for Nielsen coincidence numbers of pairs of maps (f,g): M→N between closed smooth manifolds of the same dimension. Suppose that G is a normal subgroup of Π = π1(M) with finite index and H is a normal subgroup of Δ = π1(N) with finite index such that Then we investigate the conditions for which the following averaging formula holdswhere is any pair of fixed liftings of (f, g). We prove that the averaging formula holds when M and N are orientable infra-nilmanifolds of the same dimension, and when M = N is a non-orientable infra-nilmanifold with holonomy group ℤ2 and (f, g) admits a pair of liftings on the nil-covering of M.

On weakly 𝓗-permutable subgroups of finite groups

2019 ◽  
Vol 69 (4) ◽  
pp. 763-772
Author(s):  
Chenchen Cao ◽  
Venus Amjid ◽  
Chi Zhang
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
Permutable Subgroups ◽  
A Finite Group ◽  
New Criteria

Abstract Let σ = {σi ∣i ∈ I} be some partition of the set of all primes ℙ, G be a finite group and σ(G) = {σi∣σi ∩ π(G) ≠ ∅}. G is said to be σ-primary if ∣σ(G)∣ ≤ 1. A subgroup H of G is said to be σ-subnormal in G if there exists a subgroup chain H = H0 ≤ H1 ≤ … ≤ Ht = G such that either Hi−1 is normal in Hi or Hi/(Hi−1)Hi is σ-primary for all i = 1, …, t. A set 𝓗 of subgroups of G is said to be a complete Hall σ-set of G if every non-identity 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 σi ∈ 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 all A ∈ 𝓗. We say that a subgroup H of G is weakly 𝓗-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ H𝓗, where H𝓗 is the subgroup of H generated by all those subgroups of H which are 𝓗-permutable. By using the weakly 𝓗-permutable subgroups, we establish some new criteria for a group G to be σ-soluble and supersoluble, and we also give the conditions under which a normal subgroup of G is hypercyclically embedded.

scholarly journals On groups with all subgroups almost subnormal

2004 ◽  
Vol 77 (2) ◽  
pp. 165-174 ◽  
Author(s):  
Eloisa Detomi
Keyword(s):  
Normal Subgroup ◽  
Nilpotent Group ◽  
Finite Index ◽  
Normal Closure ◽  
Fixed Integer ◽  
Torsion Free

AbstractIn this paper we consider groups in which every subgroup has finite index in the nth term of its normal closure series, for a fixed integer n. We prove that such a group is the extension of a finite normal subgroup by a nilpotent group, whose class is bounded in terms of n only, provided it is either periodic or torsion-free.

A NOTE ON GROUPS WHOSE PROPER LARGE SUBGROUPS HAVE A TRANSITIVE NORMALITY RELATION

2016 ◽  
Vol 95 (1) ◽  
pp. 38-47 ◽  
Author(s):  
FRANCESCO DE GIOVANNI ◽  
MARCO TROMBETTI
Keyword(s):  
Soluble Group ◽  
Subnormal Subgroup ◽  
Normal Subgroups ◽  
Transitive Relation ◽  
Formula Group ◽  
Subnormal Subgroups

A group $G$ is said to have the $T$-property (or to be a $T$-group) if all its subnormal subgroups are normal, that is, if normality in $G$ is a transitive relation. The aim of this paper is to investigate the behaviour of uncountable groups of cardinality $\aleph$ whose proper subgroups of cardinality $\aleph$ have a transitive normality relation. It is proved that such a group $G$ is a $T$-group (and all its subgroups have the same property) provided that $G$ has an ascending subnormal series with abelian factors. Moreover, it is shown that if $G$ is an uncountable soluble group of cardinality $\aleph$ whose proper normal subgroups of cardinality $\aleph$ have the $T$-property, then every subnormal subgroup of $G$ has only finitely many conjugates.

scholarly journals GIBBS MEASURES FOR SOS MODELS ON A CAYLEY TREE

2006 ◽  
Vol 09 (03) ◽  
pp. 471-488 ◽  
Author(s):  
U. A. ROZIKOV ◽  
Y. M. SUHOV
Keyword(s):  
Ising Model ◽  
Normal Subgroup ◽  
Finite Index ◽  
Cayley Tree ◽  
Nearest Neighbor ◽  
Gibbs Measures ◽  
Natural Generalization ◽  
Mirror Reflection ◽  
Sos Model ◽  
Translation Invariant

We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin values 0, 1,…, m, m ≥ 2, and zero external field, on a Cayley tree of order k (with k + 1 neighbors). The SOS model can be treated as a natural generalization of the Ising model (obtained for m = 1). We mainly assume that m = 2 (three spin values) and study translation-invariant (TI) and "splitting" (S) Gibbs measures (GMs). (Splitting GMs have a particular Markov-type property specific for a tree.) Furthermore, we focus on symmetric TISGMs, with respect to a "mirror" reflection of the spins. [For the Ising model (where m = 1), such measures are reduced to the "disordered" phase obtained for free boundary conditions, see Refs. 9, 10.] For m = 2, in the antiferromagnetic (AFM) case, a symmetric TISGM (and even a general TISGM) is unique for all temperatures. In the ferromagnetic (FM) case, for m = 2, the number of symmetric TISGMs and (and the number of general TISGMs) varies with the temperature: this gives an interesting example of phase transition. Here we identify a critical inverse temperature, [Formula: see text] such that [Formula: see text], there exists a unique symmetric TISGM μ* and [Formula: see text] there are exactly three symmetric TISGMs: [Formula: see text] (a "bottom" symmetric TISGM), [Formula: see text] (a "middle" symmetric TISGM) and [Formula: see text] (a "top" symmetric TISGM). For [Formula: see text] we also construct a continuum of distinct, symmertric SGMs which are non-TI. Our second result gives complete description of the set of periodic Gibbs measures for the SOS model on a Cayley tree. A complete description of periodic GMs means a characterisation of such measures with respect to any given normal subgroup of finite index in the representation group of the tree. We show that (i) for an FM SOS model, for any normal subgroup of finite index, each periodic SGM is in fact TI. Further, (ii) for an AFM SOS model, for any normal subgroup of finite index, each periodic SGM is either TI or has period two (i.e. is a chess-board SGM).

scholarly journals Groups of odd order in which every subnormal subgroup has defect at most two

1991 ◽  
Vol 51 (2) ◽  
pp. 331-342
Author(s):  
John Cossey
Keyword(s):  
Soluble Group ◽  
Subnormal Subgroup ◽  
Finite Soluble Group ◽  
Derived Length ◽  
Odd Order ◽  
Fitting Length

AbstractIn 1980, McCaughan and Stonehewer showed that a finite soluble group in which every subnormal subgroup has defect at most two has derived length at most nine and Fitting length at most five, and gave an example of derived length five and Fitting length four. In 1984 Casolo showed that derived length five and Fitting length four are best possible bounds.In this paper we show that for groups of odd order the bounds can be improved. A group of odd order with every subnormal subgroup of defect at most two has derived and Fitting length at most three, and these bounds are best possible.

scholarly journals Finite groups with given systems of generalised σ-permutable subgroups

2021 ◽  
pp. 25-33
Author(s):  
Viktoria S. Zakrevskaya
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
Chief Factor ◽  
Permutable Subgroup ◽  
Permutable Subgroups ◽  
Group A ◽  
A Finite Group

Let σ = {σi|i ∈ I } be a partition of the set of all primes ℙ and G be a finite group. A set ℋ  of subgroups of G is said to be a complete Hall σ-set of G if every member ≠1 of ℋ  is a Hall σi-subgroup of G for some i ∈ I and ℋ contains exactly one Hall σi-subgroup of G for every i such that σi ⌒ π(G)  ≠ ∅.  A group is said to be σ-primary if it is a finite σi-group for some i. A subgroup A of G is said to be: σ-permutable in G if G possesses a complete Hall σ-set ℋ  such that AH x = H  xA for all H ∈ ℋ  and all x ∈ G; σ-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤ … ≤ At = G such that either Ai − 1 ⊴ Ai or Ai /(Ai − 1)Ai is σ-primary for all i = 1, …, t; 𝔄-normal in G if every chief factor of G between AG and AG is cyclic. We say that a subgroup H of G is: (i) partially σ-permutable in G if there are a 𝔄-normal subgroup A and a σ-permutable subgroup B of G such that H = < A, B >; (ii) (𝔄, σ)-embedded in G if there are a partially σ-permutable subgroup S and a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ S ≤ H. We study G assuming that some subgroups of G are partially σ-permutable or (𝔄, σ)-embedded in G. Some known results are generalised.

A Conjecture of Bachmuth and Mochizuki on Automorphisms of Soluble Groups

1976 ◽  
Vol 28 (6) ◽  
pp. 1302-1310 ◽  
Author(s):  
Brian Hartley
Keyword(s):  
Finite Index ◽  
Soluble Group ◽  
Abelian Groups ◽  
Free Subgroup ◽  
Linear Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Group Of Automorphisms ◽  
Finitely Generated Group ◽  
Soluble Subgroup

In [1], Bachmuth and Mochizuki conjecture, by analogy with a celebrated result of Tits on linear groups [8], that a finitely generated group of automorphisms of a finitely generated soluble group either contains a soluble subgroup of finite index (which may of course be taken to be normal) or contains a non-abelian free subgroup. They point out that their conjecture holds for nilpotent-by-abelian groups and in some other cases.

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