Fundamental Groups of Graphs of Cyclic Subgroup Separable and Weakly Potent Groups

2021 ◽  
Vol 28 (01) ◽  
pp. 119-130
Author(s):  
M.S.M. Asri ◽  
K.B. Wong ◽  
P.C. Wong
Keyword(s):  
Fundamental Group ◽  
Finite Groups ◽  
Cyclic Subgroup ◽  
Infinite Order ◽  
Fundamental Groups ◽  
Subgroup Separability

We give a characterization of the cyclic subgroup separability and weak potency of the fundamental group of a graph of polycyclic-by-finite groups and free-by-finite groups amalgamating edge subgroups of the form [Formula: see text], where [Formula: see text] has infinite order and [Formula: see text] is finite.

A study of enhanced power graphs of finite groups

2019 ◽  
Vol 19 (04) ◽  
pp. 2050062 ◽  
Author(s):  
Samir Zahirović ◽  
Ivica Bošnjak ◽  
Rozália Madarász
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Cyclic Subgroup ◽  
Nilpotent Groups ◽  
Sufficient Condition ◽  
Finite Abelian Groups ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

scholarly journals Unitarization of Loop Group Representations of Fundamental Groups

2007 ◽  
Vol 187 ◽  
pp. 1-33 ◽  
Author(s):  
Josef Dorfmeister ◽  
Hongyou Wu
Keyword(s):  
Fundamental Group ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Loop Group ◽  
Fundamental Groups ◽  
Matrix Functions ◽  
Group Representations ◽  
Constant Mean Curvature Surfaces ◽  
Finite Set

AbstractIn this paper, we give a characterization of the simultaneous unitarizability of any finite set of SL(2, ℂ)-valued functions on and determine all possible ways of the unitarization. Such matrix functions can be regarded as images of the generators for the fundamental group of a surface in an -family, and the results of this paper have applications in the construction of constant mean curvature surfaces in space.

scholarly journals On the cyclic subgroup separability of the free product of two groups with commuting subgroups

2014 ◽  
Vol 24 (05) ◽  
pp. 741-756 ◽  
Author(s):  
E. V. Sokolov
Keyword(s):  
Free Product ◽  
Finite Groups ◽  
Cyclic Subgroup ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Cyclic Subgroups ◽  
Subgroup Separability ◽  
Free Product Of Groups ◽  
Product Of Groups

Let G be the free product of groups A and B with commuting subgroups H ≤ A and K ≤ B, and let 𝒞 be the class of all finite groups or the class of all finite p-groups. We derive the description of all 𝒞-separable cyclic subgroups of G provided this group is residually a 𝒞-group. We prove, in particular, that if A, B are finitely generated nilpotent groups and H, K are p′-isolated in the free factors, then all p′-isolated cyclic subgroups of G are separable in the class of all finite p-groups. The same statement is true provided A, B are free and H, K are p′-isolated and cyclic.

scholarly journals On Fuzzy Fundamental Groups and Fuzzy Folding of Fuzzy Minkowski Space

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Adeeb G. Talafha ◽  
Sahar M. Alqaraleh ◽  
A. E. El-Ahmady ◽  
Jehad Al Jaraden
Keyword(s):  
Fundamental Group ◽  
Minkowski Space ◽  
Fundamental Groups ◽  
Combinatorial Characterization ◽  
Identity Group ◽  
Fuzzy Identity

In this paper, we studied the relations between new types of fuzzy retractions, fuzzy foldings, and fuzzy deformation retractions, on fuzzy fundamental groups of the fuzzy Minkowski space M ˜ 4 . These geometrical transformations are used to give a combinatorial characterization of the fundamental groups of fuzzy submanifolds on M ˜ 4 . Then, the fuzzy fundamental groups of the fuzzy geodesics and the limit fuzzy foldings of M ˜ 4 are presented and obtained. Finally, we proved a sequence of theorems concerning the isomorphism between the fuzzy fundamental group and the fuzzy identity group.

Characterization of small polygroups by their fundamental groups

2019 ◽  
Vol 11 (04) ◽  
pp. 1950047 ◽  
Author(s):  
D. Heidari ◽  
B. Davvaz
Keyword(s):  
Fundamental Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Fundamental Groups ◽  
Trivial Group ◽  
Necessary And Sufficient

In this paper, we consider polygroup [Formula: see text] and prove necessary and sufficient conditions such that [Formula: see text] is non-commutative. Then by using the Maple programming, we obtain all polygroups of order less than five up to isomorphism. In fact, we determine all 115 non-isomorphic polygroups of order less than five and characterize them by their fundamental groups, i.e., polygroups with same fundamental group, say [Formula: see text], classifies in the class [Formula: see text]. Finally, we obtain that the fundamental groups of 94 polygroups are the trivial group. The numbers of polygroups in classes [Formula: see text] and [Formula: see text] are 16 and 3, respectively, and the classes [Formula: see text] and [Formula: see text] are singleton.

Subgroup Separability of Generalized Free Products of Free-By-Finite Groups

1993 ◽  
Vol 36 (4) ◽  
pp. 385-389 ◽  
Author(s):  
R. B. J. T. Allenby ◽  
C. Y. Tang
Keyword(s):  
Finite Groups ◽  
Word Problems ◽  
Cyclic Subgroup ◽  
Finitely Generated ◽  
Free Products ◽  
Subgroup Separability ◽  
Solvable Word ◽  
Generalized Free Products

AbstractWe prove that generalized free products of finitely generated free-byfinite groups amalgamating a cyclic subgroup are subgroup separable. From this it follows that if where t ≥ 1 and u, v are words on {a1,...,am} and {b1,...,bn} respectively then G is subgroup separable thus generalizing a result in [9] that such groups have solvable word problems.

scholarly journals The Cyclic Subgroup Separability of Certain Generalized Free Products of Two Groups

2010 ◽  
Vol 17 (04) ◽  
pp. 577-582 ◽  
Author(s):  
P. A. Bobrovskii ◽  
E. V. Sokolov
Keyword(s):  
Finite Groups ◽  
Cyclic Subgroup ◽  
Free Products ◽  
Residually Finite ◽  
Residually Finite Groups ◽  
Subgroup Separability ◽  
Generalized Free Products

Free products of two residually finite groups with amalgamated retracts are considered. It is proved that a cyclic subgroup of such a group is not finitely separable if, and only if, it is conjugated with a subgroup of a free factor which is not finitely separable in this factor. A similar result is obtained for the case of separability in the class of finite p-groups.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

scholarly journals Polygonal products of polycyclic by finite groups

1996 ◽  
Vol 54 (3) ◽  
pp. 369-372 ◽  
Author(s):  
R.B.J.T. Allenby
Keyword(s):  
Finite Groups ◽  
Cyclic Subgroup ◽  
Substantial Improvement ◽  
Normal Subgroups ◽  
Recent Theorem

We prove that a polygonal product of polycyclic by finite groups amalgamating normal subgroups, with trivial mutual intersections, is cyclic subgroup separable. Because of a recent example (stated below) of the author this substantial improvement on a recent theorem of Kim is essentially best possible.

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