scholarly journals On the shellability of the order complex of the subgroup lattice of a finite group

2001 ◽  
Vol 353 (7) ◽  
pp. 2689-2703 ◽  
Author(s):  
John Shareshian
Keyword(s):  
Finite Group ◽  
Subgroup Lattice ◽  
Order Complex ◽  
A Finite Group

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 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 Permutability degrees of finite groups

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2165-2175
Author(s):  
D.E. Otera ◽  
F.G. Russo
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Subgroup Lattice ◽  
Cyclic Subgroups ◽  
A Finite Group

Given a finite group G, we introduce the permutability degree of G, as pd(G) = 1/|G| |L(G)| ?X?L(G)|PG(X)|, where L(G) is the subgroup lattice of G and PG(X) the permutizer of the subgroup X in G, that is, the subgroup generated by all cyclic subgroups of G that permute with X ? L(G). The number pd(G) allows us to find some structural restrictions on G. Successively, we investigate the relations between pd(G), the probability of commuting subgroups sd(G) of G and the probability of commuting elements d(G) of G. Proving some inequalities between pd(G), sd(G) and d(G), we correlate these notions.

Subnormality and the Chermak–Delgado lattice

2019 ◽  
Vol 19 (08) ◽  
pp. 2050141
Author(s):  
W. Cocke
Keyword(s):  
Finite Group ◽  
Height Function ◽  
Subgroup Lattice ◽  
A Finite Group

The Chermak–Delgado lattice of a finite group [Formula: see text] is a sublattice of the subgroup lattice of [Formula: see text] that has attracted interest since its discovery. In this paper, we show that every subgroup of [Formula: see text] in the Chermak–Delgado lattice is subnormal in [Formula: see text] with subnormal depth bounded by both the depth and height function of the Chermak–Delgado lattice; we provide a nontrivial example showing that our bounds are sharp. We also show that determining whether a given subgroup [Formula: see text] is in the Chermak–Delgado lattice can be decided by examining only those subgroups of [Formula: see text] that are comparable with [Formula: see text].

ON HIGHER FROBENIUS–SCHUR INDICATORS

2021 ◽  
pp. 1-11
Author(s):  
YANJUN LIU ◽  
WOLFGANG WILLEMS
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Irreducible Characters ◽  
A Finite Group

Abstract Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.

scholarly journals Finite groups with some weakly pronormal subgroups

2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

scholarly journals Random additions in urns of integers

2021 ◽  
Vol 58 (2) ◽  
pp. 335-346
Author(s):  
Mackenzie Simper
Keyword(s):  
Finite Group ◽  
Random Process ◽  
Uniform Distribution ◽  
Discrete Time ◽  
Random Variable ◽  
Urn Model ◽  
Infinite Type ◽  
The Mean ◽  
A Finite Group ◽  
Proof Of Convergence

AbstractConsider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir (2005) Ann. Prob.33, 2036 for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.

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