Locally soluble-by-finite groups with small deviation for non-subnormal subgroups
نویسندگان
چکیده
A group G has subnormal deviation at most 1 if, for every descending chain H0 > H1 > . . . of non-subnormal subgroups of G, for all but finitely many i there is no infinite descending chain of non-subnormal subgroups of G that contain Hi+1 and are contained in Hi. This property P, say, was investigated in a previous paper by the authors, where soluble groups with P and locally nilpotent groups with Pwere effectively classified. The present article affirms a conjecture from that article by showing that locally soluble-by-finite groups with P are soluble-by-finite and are therefore classified.
منابع مشابه
Groups with the weak minimal condition for non-subnormal subgroups II
Let G be a group with the property that there are no infinite descending chains of non-subnormal subgroups of G for which all successive indices are infinite. The main result is that if G is a locally (soluble-by-finite) group with this property then either G has all subgroups subnormal or G is a soluble-by-finite minimax group. This result fills a gap left in an earlier paper by the same autho...
متن کاملFinite groups in which normality, permutability or Sylow permutability is transitive
Y. Li gave a characterization of the class of finite soluble groups in which every subnormal subgroup is normal by means of NE -subgroups: a subgroup H of a group G is called an NE -subgroup of G if NG(H) ∩ H = H. We obtain a new characterization of these groups related to the local Wielandt subgroup. We also give characterizations of the classes of finite soluble groups in which every subnorma...
متن کاملGroups with countably many subgroups
We describe soluble groups in which the set of all subgroups is countable and show that locally (soluble-byfinite) groups with this property are soluble-by-finite. Further, we construct a nilpotent group with uncountably many subgroups in which the set of all abelian subgroups is countable.
متن کاملON p-NILPOTENCY OF FINITE GROUPS WITH SS-NORMAL SUBGROUPS
Abstract. A subgroup H of a group G is said to be SS-embedded in G if there exists a normal subgroup T of G such that HT is subnormal in G and H T H sG , where H sG is the maximal s- permutable subgroup of G contained in H. We say that a subgroup H is an SS-normal subgroup in G if there exists a normal subgroup T of G such that G = HT and H T H SS , where H SS is an SS-embedded subgroup of ...
متن کاملGroups with one conjugacy class of non-normal subgroups - a short proof
For a finite group $G$ let $nu(G)$ denote the number of conjugacy classes of non-normal subgroups of $G$. We give a short proof of a theorem of Brandl, which classifies finite groups with $nu(G)=1$.
متن کامل