scholarly journals Groups whose Chermak–Delgado lattice is a subgroup lattice of an elementary abelian p-group

2021 ◽  
pp. 1-8
Author(s):  
Lijian An
Keyword(s):  
Subgroup Lattice

scholarly journals A lattice-theoretic characterization of pure subgroups of Abelian groups

2021 ◽  
Author(s):  
M. Ferrara ◽  
M. Trombetti
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Subgroup Lattice ◽  
General Definition ◽  
Short Paper ◽  
Theoretic Characterization ◽  
Definition Of

AbstractLet G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice $$\mathcal {L}(H)$$ L ( H ) embeds in $$\mathcal {L}(G)$$ L ( G ) . It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.

The intrinsic group–subgroup structures of the Diamond and Gyroid minimal surfaces in their conventional unit cells

2022 ◽  
Vol 78 (1) ◽  
Author(s):  
Martin Cramer Pedersen ◽  
Vanessa Robins ◽  
Stephen T. Hyde
Keyword(s):  
Minimal Surfaces ◽  
Subgroup Lattice ◽  
Unit Cells ◽  
Lattice Graphs ◽  
Conventional Unit

The intrinsic, hyperbolic crystallography of the Diamond and Gyroid minimal surfaces in their conventional unit cells is introduced and analysed. Tables are constructed of symmetry subgroups commensurate with the translational symmetries of the surfaces as well as group–subgroup lattice graphs.

On two sublattices of the subgroup lattice of a finite group

2019 ◽  
Vol 22 (6) ◽  
pp. 1035-1047 ◽  
Author(s):  
Zhang Chi ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Subnormal Subgroup ◽  
Subgroup Lattice ◽  
Chief Factor ◽  
A Finite Group ◽  
Fitting Formation

Abstract Let {\mathfrak{F}} be a non-empty class of groups, let G be a finite group and let {\mathcal{L}(G)} be the lattice of all subgroups of G. A chief {H/K} factor of G is {\mathfrak{F}} -central in G if {(H/K)\rtimes(G/C_{G}(H/K))\in\mathfrak{F}} . Let {\mathcal{L}_{c\mathfrak{F}}(G)} be the set of all subgroups A of G such that every chief factor {H/K} of G between {A_{G}} and {A^{G}} is {\mathfrak{F}} -central in G; {\mathcal{L}_{\mathfrak{F}}(G)} denotes the set of all subgroups A of G with {A^{G}/A_{G}\in\mathfrak{F}} . We prove that the set {\mathcal{L}_{c\mathfrak{F}}(G)} and, in the case when {\mathfrak{F}} is a Fitting formation, the set {\mathcal{L}_{\mathfrak{F}}(G)} are sublattices of the lattice {\mathcal{L}(G)} . We also study conditions under which the lattice {\mathcal{L}_{c\mathfrak{N}}(G)} and the lattice of all subnormal subgroup of G are modular.

scholarly journals A note on growth sequences of finite simple groups

1995 ◽  
Vol 51 (3) ◽  
pp. 495-499 ◽  
Author(s):  
Ahmad Erfanian ◽  
James Wiegold
Keyword(s):  
Simple Group ◽  
Subgroup Lattice ◽  
Möbius Function ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Mobius Function ◽  
New Formula

The aim of this paper is to give a new precise formula for h(n, A), where A is a finite non-abelian simple group, h(n, A) is the maximum number such that Ah(n, A) can ke generated by n elements, and n ≥ 2. P. Hall gave a formula for h(n, A) in terms of the Möbius function of the subgroup lattice of A; the new formula involves a concept called cospread associated with that of spread as explained in Brenner and Wiegold (1975).

scholarly journals A connection between the number of subgroups and the order of a finite group

2020 ◽  
pp. 2250001
Author(s):  
Mihai-Silviu Lazorec
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Finite Abelian Group ◽  
Subgroup Lattice ◽  
Minimum Value ◽  
A Finite Group

For a finite group [Formula: see text], we associate the quantity [Formula: see text], where [Formula: see text] is the subgroup lattice of [Formula: see text]. Different properties and problems related to this ratio are studied throughout this paper. We determine the second minimum value of [Formula: see text] on the class of [Formula: see text]-groups of order [Formula: see text], where [Formula: see text] is an integer. We show that the set containing the quantities [Formula: see text], where [Formula: see text] is a finite (abelian) group, is dense in [Formula: see text] Finally, we consider [Formula: see text] to be a function on [Formula: see text] and we indicate some of its properties, the main result being the classification of finite abelian [Formula: see text]-groups [Formula: see text] satisfying [Formula: see text]

scholarly journals Detecting the index of a subgroup in the subgroup lattice

2004 ◽  
Vol 133 (4) ◽  
pp. 979-985 ◽  
Author(s):  
M. De Falco ◽  
F. de Giovanni ◽  
C. Musella ◽  
R. Schmidt
Keyword(s):  
Subgroup Lattice

scholarly journals On the number of diamonds in the subgroup lattice of a finite abelian group

2016 ◽  
Vol 24 (2) ◽  
pp. 205-215
Author(s):  
Dan Gregorian Fodor ◽  
Marius Tărnăuceanu
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Subgroup Lattice ◽  
Current Paper ◽  
Explicit Formulas ◽  
Counting Problem

Abstract The main goal of the current paper is to determine the total number of diamonds in the subgroup lattice of a finite abelian group. This counting problem is reduced to finite p-groups. Explicit formulas are obtained in some particular cases.

scholarly journals On some classes of sublattices of the subgroup lattice

2019 ◽  
pp. 35-47 ◽  
Author(s):  
Alexander N. Skiba
Keyword(s):  
Normal Subgroup ◽  
Factor Group ◽  
Subgroup Lattice ◽  
Normal Closure ◽  
Normal Subgroups ◽  
Normal Section ◽  
Normal Core

In this paper G always denotes a group. If K and H are subgroups of G, where K is a normal subgroup of H, then the factor group of H by K is called a section of G. Such a section is called normal, if K and H are normal subgroups of G, and trivial, if K and H are equal. We call any set S of normal sections of G a stratification of G, if S contains every trivial normal section of G, and we say that a stratification S of G is G-closed, if S contains every such a normal section of G, which is G-isomorphic to some normal section of G belonging S. Now let S be any G-closed stratification of G, and let L be the set of all subgroups A of G such that the factor group of V by W, where V is the normal closure of A in G and W is the normal core of A in G, belongs to S. In this paper we describe the conditions on S under which the set L is a sublattice of the lattice of all subgroups of G and we also discuss some applications of this sublattice in the theory of generalized finite T-groups.

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