Chief Factor Sizes in Finitely Generated Varieties

K. A. Kearnes; E. W. Kiss; Á. Szendrei; R. D. Willard

doi:10.4153/cjm-2002-028-x

Chief Factor Sizes in Finitely Generated Varieties

Canadian Journal of Mathematics ◽

10.4153/cjm-2002-028-x ◽

2002 ◽

Vol 54 (4) ◽

pp. 736-756

Author(s):

K. A. Kearnes ◽

E. W. Kiss ◽

Á. Szendrei ◽

R. D. Willard

Keyword(s):

Chief Factor ◽

Finitely Generated ◽

Finite Algebras ◽

Congruence Identity

AbstractLet A be a k-element algebra whose chief factor size is c. We show that if B is in the variety generated by A, then any abelian chief factor of B that is not strongly abelian has size at most ck−1. This solves Problem 5 of The Structure of Finite Algebras, by D. Hobby and R. McKenzie. We refine this bound to c in the situation where the variety generated by A omits type 1. As a generalization, we bound the size of multitraces of types 1, 2, and 3 by extending coordinatization theory. Finally, we exhibit some examples of bad behavior, even in varieties satisfying a congruence identity.

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ON THE COMPLEXITY OF DECIDING HOMOMORPHISM-HOMOGENEITY FOR FINITE ALGEBRAS

International Journal of Algebra and Computation ◽

10.1142/s0218196713500148 ◽

2013 ◽

Vol 23 (03) ◽

pp. 663-672 ◽

Cited By ~ 1

Author(s):

DRAGAN MAŠULOVIĆ

Keyword(s):

Fundamental Operation ◽

Finitely Generated ◽

Homogeneous Groups ◽

Finite Algebras ◽

Homogeneous Algebras

In 2006, Cameron and Nešetřil introduced the following variant of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finitely generated substructures of the structure extends to an endomorphism of the structure. In several recent papers homomorphism-homogeneous objects in some well-known classes of algebras have been described (e.g. monounary algebras and lattices), while finite homomorphism-homogeneous groups were described in 1979 under the name of finite quasi-injective groups. In this paper we show that, in general, deciding homomorphism-homogeneity for finite algebras with finitely many fundamental operations and with at least one at least binary fundamental operation is coNP-complete. Therefore, unless P = coNP, there is no feasible characterization of finite homomorphism-homogeneous algebras of this kind.

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Groups with finitely many conjugacy classes of subgroups with large subnormal defect

Glasgow Mathematical Journal ◽

10.1017/s0017089500030408 ◽

1995 ◽

Vol 37 (1) ◽

pp. 69-71 ◽

Cited By ~ 3

Author(s):

Howard Smith

Keyword(s):

Positive Integer ◽

Free Group ◽

Conjugacy Classes ◽

The Other ◽

Chief Factor ◽

Simple Groups ◽

Finitely Generated ◽

Finite Image ◽

Chief Factors ◽

Locally Graded Groups

Given a group G and a positive integer k, let vk(G) denote the number of conjugacy classes of subgroups of G which are not subnormal of defect at most k. Groups G such that vkG) < ∝ for some k are considered in Section 2 of [1], and Theorem 2.4 of that paper states that an infinite group G for which vk(G) < ∝ (for some k) is nilpotent provided only that all chief factors of G are locally (soluble or finite). Now it is easy to see that a group G whose chief factors are of this type is locally graded, that is, every nontrivial, finitely generated subgroup F of G has a nontrivial finite image (since there is a chief factor H/K of G such that F is contained in H but not in K). On the other hand, every (locally) free group is locally graded and so there is in general no restriction on the chief factors of such groups. The class of locally graded groups is a suitable class to consider if one wishes to do no more than exclude the occurrence of finitely generated, infinite simple groups and, in particular, Tarski p-groups. As pointed out in [1], Ivanov and Ol'shanskiĭ have constructed (finitely generated) infinite simple groups all of whose proper nontrivial subgroups are conjugate; clearly a group G with this property satisfies v1(G) = l. The purpose of this note is to provide the following generalization of the above-mentioned theorem from [1].

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PROFINITENESS IN FINITELY GENERATED VARIETIES IS UNDECIDABLE

Journal of Symbolic Logic ◽

10.1017/jsl.2017.89 ◽

2018 ◽

Vol 83 (04) ◽

pp. 1566-1578 ◽

Cited By ~ 1

Author(s):

ANVAR M. NURAKUNOV ◽

MICHAŁ M. STRONKOWSKI

Keyword(s):

Finite Type ◽

Finite Algebra ◽

Topological Algebra ◽

Inverse Limits ◽

Finitely Generated ◽

Finite Algebras ◽

Image Position ◽

Profinite Algebras

AbstractProfinite algebras are exactly those that are isomorphic to inverse limits of finite algebras. Such algebras are naturally equipped with Boolean topologies. A variety ${\cal V}$ is standard if every Boolean topological algebra with the algebraic reduct in ${\cal V}$ is profinite.We show that there is no algorithm which takes as input a finite algebra A of a finite type and decide whether the variety $V\left( {\bf{A}} \right)$ generated by A is standard. We also show the undecidability of some related properties. In particular, we solve a problem posed by Clark, Davey, Freese, and Jackson.We accomplish this by combining two results. The first one is Moore’s theorem saying that there is no algorithm which takes as input a finite algebra A of a finite type and decides whether $V\left( {\bf{A}} \right)$ has definable principal subcongruences. The second is our result saying that possessing definable principal subcongruences yields possessing finitely determined syntactic congruences for varieties. The latter property is known to yield standardness.

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Finite algebras that generate an injectively complete modular variety

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700029750 ◽

1991 ◽

Vol 44 (2) ◽

pp. 303-324 ◽

Cited By ~ 1

Author(s):

Keith A. Kearnes

Keyword(s):

Finite Type ◽

Finite Algebra ◽

Finitely Generated ◽

Modular Variety ◽

Finite Algebras ◽

Congruence Modular Variety ◽

Congruence Distributive

We extend Kollár's result on finitely generated, injectively complete congruence distributive varieties to the congruence modular setting. By doing so we show that, given any finite algebra A of finite type, there is an algorithm to decide whether V(A) is an injectively complete, congruence modular variety.

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Automorphisms of finite order of nilpotent groups II

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.51.2014.4.1297 ◽

2014 ◽

Vol 51 (4) ◽

pp. 547-555 ◽

Cited By ~ 1

Author(s):

B. Wehrfritz

Keyword(s):

Nilpotent Group ◽

Finite Order ◽

Finite Index ◽

Nilpotent Groups ◽

Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

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