scholarly journals ‘COHOMOLOGY AND PROFINITE TOPOLOGIES FOR SOLVABLE GROUPS OF FINITE RANK’

2015 ◽  
Vol 91 (2) ◽  
pp. 351-352
Author(s):  
KARL LORENSEN
Keyword(s):  
Solvable Group ◽  
Finite Rank ◽  
Solvable Groups ◽  
Section Rank

AbstractWe remedy an omission in the proof of Proposition 2.7 of the paper ‘Cohomology and profinite topologies for solvable groups of finite rank’, Bull. Aust. Math. Soc.86 (2012), 254–265. This proposition states that a solvable group with finite abelian section rank has merely finitely many subgroups of any given index.

A Finite Index Property of Certain Solvable Groups

1984 ◽  
Vol 27 (4) ◽  
pp. 485-489
Author(s):  
A. H. Rhemtulla ◽  
H. Smith
Keyword(s):  
Solvable Group ◽  
Finite Index ◽  
Finite Rank ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Free Solvable Group ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
Torsion Free

AbstractA group G is said to have the FINITE INDEX property (G is an FI-group) if, whenever H≤G, xp ∈ H for some x in G and p > 0, then |〈H, x〉: H| is finite. Following a brief discussion of some locally nilpotent groups with this property, it is shown that torsion-free solvable groups of finite rank which have the isolator property are FI-groups. It is deduced from this that a finitely generated torsion-free solvable group has an FI-subgroup of finite index if and only if it has finite rank.

scholarly journals COHOMOLOGY AND PROFINITE TOPOLOGIES FOR SOLVABLE GROUPS OF FINITE RANK

2012 ◽  
Vol 86 (2) ◽  
pp. 254-265 ◽  
Author(s):  
KARL LORENSEN
Keyword(s):  
Normal Subgroup ◽  
Solvable Group ◽  
Finite Index ◽  
Finite Rank ◽  
Solvable Groups ◽  
Graphic Module

AbstractAssume that G is a solvable group whose elementary abelian sections are all finite. Suppose, further, that p is a prime such that G fails to contain any subgroups isomorphic to Cp∞. We show that if G is nilpotent, then the pro-p completion map $G\to \hat {G}_p$ induces an isomorphism $H^\ast (\hat {G}_p,M)\to H^\ast (G,M)$ for any discrete $\hat {G}_p$-module M of finite p-power order. For the general case, we prove that G contains a normal subgroup N of finite index such that the map $H^\ast (\hat {N}_p,M)\to H^\ast (N,M)$ is an isomorphism for any discrete $\hat {N}_p$-module M of finite p-power order. Moreover, if G lacks any Cp∞-sections, the subgroup N enjoys some additional special properties with respect to its pro-p topology.

𝒫-characters and the structure of finite solvable groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiakuan Lu ◽  
Kaisun Wu ◽  
Wei Meng
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Irreducible Constituent ◽  
A Finite Group

AbstractLet 𝐺 be a finite group. An irreducible character of 𝐺 is called a 𝒫-character if it is an irreducible constituent of (1_{H})^{G} for some maximal subgroup 𝐻 of 𝐺. In this paper, we obtain some conditions for a solvable group 𝐺 to be 𝑝-nilpotent or 𝑝-closed in terms of 𝒫-characters.

On normal subgroups of M-groups

2019 ◽  
Vol 18 (04) ◽  
pp. 1950074
Author(s):  
Xuewu Chang
Keyword(s):  
Normal Subgroup ◽  
Solvable Group ◽  
Solvable Groups ◽  
Hall Subgroup ◽  
Embedding Theorem ◽  
The Other ◽  
Finite Solvable Group ◽  
Embedding Problem ◽  
Normal Subgroups ◽  
Other Hand

The normal embedding problem of finite solvable groups into [Formula: see text]-groups was studied. It was proved that for a finite solvable group [Formula: see text], if [Formula: see text] has a special normal nilpotent Hall subgroup, then [Formula: see text] cannot be a normal subgroup of any [Formula: see text]-group; on the other hand, if [Formula: see text] has a maximal normal subgroup which is an [Formula: see text]-group, then [Formula: see text] can occur as a normal subgroup of an [Formula: see text]-group under some suitable conditions. The results generalize the normal embedding theorem on solvable minimal non-[Formula: see text]-groups to arbitrary [Formula: see text]-groups due to van der Waall, and also cover the famous counterexample given by Dade and van der Waall independently to the Dornhoff’s conjecture which states that normal subgroups of arbitrary [Formula: see text]-groups must be [Formula: see text]-groups.

The Laitinen Conjecture for finite non-solvable groups

2012 ◽  
Vol 56 (1) ◽  
pp. 303-336 ◽  
Author(s):  
Krzysztof Pawałowski ◽  
Toshio Sumi
Keyword(s):  
Finite Group ◽  
Fixed Points ◽  
Solvable Group ◽  
Prime Power ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
Prime Power Order ◽  
Algebraic Condition ◽  
Smooth Action

AbstractFor any finite group G, we impose an algebraic condition, the Gnil-coset condition, and prove that any finite Oliver group G satisfying the Gnil-coset condition has a smooth action on some sphere with isolated fixed points at which the tangent G-modules are not isomorphic to each other. Moreover, we prove that, for any finite non-solvable group G not isomorphic to Aut(A6) or PΣL(2, 27), the Gnil-coset condition holds if and only if rG ≥ 2, where rG is the number of real conjugacy classes of elements of G not of prime power order. As a conclusion, the Laitinen Conjecture holds for any finite non-solvable group not isomorphic to Aut(A6).

A Note on Brauer Character Degrees of Solvable Groups

1991 ◽  
Vol 34 (3) ◽  
pp. 423-425 ◽  
Author(s):  
You-Qiang Wang
Keyword(s):  
Solvable Group ◽  
Solvable Groups ◽  
Character Degrees ◽  
Finite Solvable Group ◽  
Prime Integer ◽  
Derived Length ◽  
Brauer Character ◽  
Brauer Characters

AbstractLet G be a finite solvable group. Fix a prime integer p and let t be the number of distinct degrees of irreducible Brauer characters of G with respect to the prime p. We obtain the bound 3t — 2 for the derived length of a Hall p'-subgroup of G. Furthermore, if |G| is odd, then the derived length of a Hall p'-subgroup of G is bounded by /.

On the profinite topology on solvable groups

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Khadijeh Alibabaei
Keyword(s):  
Abelian Group ◽  
Wreath Product ◽  
Solvable Group ◽  
Solvable Groups ◽  
Polycyclic Group ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Finitely Generated Group ◽  
Hnn Extension ◽  
Profinite Topology

AbstractWe show that the wreath product of a finitely generated abelian group with a polycyclic group is a LERF group. This theorem yields as a corollary that finitely generated free metabelian groups are LERF, a result due to Coulbois. We also show that a free solvable group of class 3 and rank at least 2 does not contain a strictly ascending HNN-extension of a finitely generated group. Since such groups are known not to be LERF, this settles, in the negative, a question of J. O. Button.

Character correspondences in solvable groups with a self-normalizing sylow subgroup

2019 ◽  
Vol 19 (10) ◽  
pp. 2050190
Author(s):  
Carolina Vallejo Rodríguez
Keyword(s):  
Solvable Group ◽  
Irreducible Character ◽  
Sylow Subgroup ◽  
Solvable Groups ◽  
Character Formula ◽  
Finite Solvable Group ◽  
Irreducible Characters ◽  
Made In

Let [Formula: see text] be a finite solvable group and let [Formula: see text] for some prime [Formula: see text]. Whenever [Formula: see text] is odd, Isaacs described a correspondence between irreducible characters of degree not divisible by [Formula: see text] of [Formula: see text] and [Formula: see text]. This correspondence is natural in the sense that an algorithm is provided to compute it, and the result of the application of the algorithm does not depend on choices made. In the case where [Formula: see text], G. Navarro showed that every irreducible character [Formula: see text] of degree not divisible by [Formula: see text] has a unique linear constituent [Formula: see text] when restricted to [Formula: see text], and that the map [Formula: see text] defines a bijection. Navarro’s bijection is obviously natural in the sense described above. We show that these two correspondences are the same under the intersection of the hypotheses.

scholarly journals Locally defined Fitting classes

1975 ◽  
Vol 20 (1) ◽  
pp. 25-32 ◽  
Author(s):  
Patrick D' Arcy
Keyword(s):  
Conjugacy Class ◽  
Solvable Group ◽  
Solvable Groups ◽  
Fitting Class ◽  
Finite Solvable Group ◽  
Saturated Formations ◽  
Definition Of ◽  
Image Position ◽  
Fitting Classes

Fitting classes of finite solvable groups were first considered by Fischer, who with Gäschutz and Hartley (1967) showed in that in each finite solvable group there is a unique conjugacy class of “-injectors”, for a Fitting class. In general the behaviour of Fitting classes and injectors seems somewhat mysterious and hard to determine. This is in contrast to the situation for saturated formations and -projectors of finite solvable groups which, because of the equivalence saturated formations and locally defined formations, can be studied in a much more detailed way. However for those Fitting classes that are “locally defined” the theory of -injectors can be made more explicit by considering various centralizers involving the local definition of , giving results analogous to some of those concerning locally defined formations. Particular attention will be given to the subgroup B() defined by where the set {(p)} of Fitting classes locally defines , and the Sp are the Sylow p-subgroups associated with a given Sylow system − B() plays a role very much like that of Graddon's -reducer in Graddon (1971). An -injector of B() is an -injector of G, and for certain simple B() is an -injector of G.

Locally solvable groups of finite rank. II

1969 ◽  
Vol 8 (6) ◽  
pp. 391-393 ◽  
Author(s):  
Yu. I. Merzlyakov
Keyword(s):  
Finite Rank ◽  
Solvable Groups
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