Retracts and Injectives

1982 ◽  
Vol 25 (4) ◽  
pp. 462-467 ◽  
Author(s):  
Shalom Feigelstock ◽  
Aaron Klein
Keyword(s):  
Polynomial Identity ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Embedding Theorems

AbstractEmbedding theorems are employed to show that many important categories do not possess non-trivial retracts or injectives. E.g., the categories of monoids, groups, rings, rings with unity, polynomial identity rings, nilpotent groups, solvable groups, and several varieties of groups.

scholarly journals Extensions of finite nilpotent groups

1970 ◽  
Vol 2 (2) ◽  
pp. 267-274
Author(s):  
John Poland
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Theoretic Property ◽  
Joint Paper ◽  
The Core ◽  
A Finite Group

If G is a finite group and P is a group-theoretic property, G will be called P-max-core if for every maximal subgroup M of G, M/MG has property P where MG = ∩ is the core of M in G. In a joint paper with John D. Dixon and A.H. Rhemtulla, we showed that if p is an odd prime and G is (p-nilpotent)-max-core, then G is p-solvable, and then using the techniques of the theory of solvable groups, we characterized nilpotent-max-core groups as finite nilpotent-by-nilpotent groups. The proof of the first result used John G. Thompson's p-nilpotency criterion and hence required p > 2. In this paper I show that supersolvable-max-core groups (and hence (2-nilpotent)-max-core groups) need not be 2-solvable (that is, solvable). Also I generalize the second result, among others, and characterize (p-nilpotent)-max-core groups (for p an odd prime) as finite nilpotent-by-(p-nilpotent) groups.

Embedding theorems for nilpotent groups

1973 ◽  
Vol 13 (4) ◽  
pp. 597-603 ◽  
Author(s):  
V. A. Roman'kov
Keyword(s):  
Nilpotent Groups ◽  
Embedding Theorems

Finite Subgroups in Integral Group Rings

1996 ◽  
Vol 48 (6) ◽  
pp. 1170-1179 ◽  
Author(s):  
Michael A. Dokuchaev ◽  
Stanley O. Juriaans
Keyword(s):  
Solvable Groups ◽  
Nilpotent Groups ◽  
Fourth Power ◽  
Group Rings ◽  
Integral Group ◽  
Frobenius Groups ◽  
Integral Group Rings ◽  
Finite Subgroups ◽  
Generalized Quaternion

AbstractA p-subgroup version of the conjecture of Zassenhaus is proved for some finite solvable groups including solvable groups in which any Sylow p-subgroup is either abelian or generalized quaternion, solvable Frobenius groups, nilpotent-by-nilpotent groups and solvable groups whose orders are not divisible by the fourth power of any prime.

On permutable strongly 2-maximal and strongly 3-maximal subgroups

2021 ◽  
pp. 121-129
Author(s):  
Yuliya V. Gorbatova
Keyword(s):  
Nilpotent Group ◽  
Solvable Group ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Maximal Subgroups ◽  
Finite Solvable Group ◽  
Prime Divisors ◽  
Sylow Subgroups ◽  
Schmidt Group ◽  
Number Of Prime Divisors

We describe the structure of finite solvable non-nilpotent groups in which every two strongly n-maximal subgroups are permutable (n = 2; 3). In particular, it is shown for a solvable non-nilpotent group G that any two strongly 2-maximal subgroups are permutable if and only if G is a Schmidt group with Abelian Sylow subgroups. We also prove the equivalence of the structure of non-nilpotent solvable groups with permutable 3-maximal subgroups and with permutable strongly 3-maximal subgroups. The last result allows us to classify all finite solvable groups with permutable strongly 3-maximal subgroups, and we describe 14 classes of groups with this property. The obtained results also prove the nilpotency of a finite solvable group with permutable strongly n -maximal subgroups if the number of prime divisors of the order of this group strictly exceeds n (n=2; 3).

scholarly journals The complexity of the equation solvability problem over semipattern groups

2017 ◽  
Vol 27 (02) ◽  
pp. 259-272 ◽  
Author(s):  
Attila Földvári
Keyword(s):  
Original Problem ◽  
Main Idea ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Matrix Groups ◽  
Semidirect Products ◽  
Solvability Problem ◽  
Equation Solvability

The complexity of the equation solvability problem is known for nilpotent groups, for not solvable groups and for some semidirect products of Abelian groups. We provide a new polynomial time algorithm for deciding the equation solvability problem over certain semidirect products, where the first factor is not necessarily Abelian. Our main idea is to represent such groups as matrix groups, and reduce the original problem to equation solvability over the underlying field. Further, we apply this new method to give a much more efficient algorithm for equation solvability over nilpotent rings than previously existed.

scholarly journals On Properties Possessed by Solvable and Nilpotent Groups

1969 ◽  
Vol 9 (1-2) ◽  
pp. 218-227 ◽  
Author(s):  
Christine Ayoub
Keyword(s):  
Solvable Groups ◽  
Nilpotent Groups ◽  
Invariant System ◽  
Central System

The object of this note is to study two properties of groups, which we will denote by (*) and (**). The property (*) is possessed by solvable groups (and in fact, by groups which have a solvable invariant system) and the property (**) is possessed by nilpotent groups (and in fact, by groups which have a central system).

scholarly journals Model theoretic connected components of finitely generated nilpotent groups

2013 ◽  
Vol 78 (1) ◽  
pp. 245-259 ◽  
Author(s):  
Nathan Bowler ◽  
Cong Chen ◽  
Jakub Gismatullin
Keyword(s):  
Nilpotent Group ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Connected Components ◽  
Connected Component ◽  
Finitely Generated ◽  
Image Position

AbstractWe prove that for a finitely generated infinite nilpotent group G with structure (G, ·, …), the connected component G*0 of a sufficiently saturated extension G* of G exists and equalsWe construct an expansion of ℤ by a predicate (ℤ, +, P) such that the type-connected component is strictly smaller than ℤ*0. We generalize this to finitely generated virtually solvable groups. As a corollary of our construction we obtain an optimality result for the van der Waerden theorem for finite partitions of groups.

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.

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