# On an Inequality of Dimension-like Invariants for Finite Groups

@article{Fernando2015OnAI, title={On an Inequality of Dimension-like Invariants for Finite Groups}, author={Ravi Fernando}, journal={arXiv: Group Theory}, year={2015} }

In this paper, we introduce several notions of "dimension" of a finite group, involving sizes of generating sets and certain configurations of maximal subgroups. We focus on the inequality $m(G) \leq \mathrm{MaxDim}(G)$, giving a family of examples where the inequality is strict, and showing that equality holds if $G$ is supersolvable.

## 10 Citations

### On Behaviors of Maximal Dimension

- Mathematics
- 2018

In this paper, we investigate behaviors of Maximal Dimension, a group invariant involving certain configuration of maximal subgroups, which we denote by MaxDim. We prove that in some special cases,…

### On the dimension of an Abelian group

- MathematicsNotes on Number Theory and Discrete Mathematics
- 2021

We introduce a measure of dimensionality of an Abelian group. Our definition of dimension is based on studying perpendicularity relations in an Abelian group. For G ≅ ℤn, dimension and rank coincide…

### On the Intersection Numbers of Finite Groups

- Mathematics
- 2019

The covering number of a nontrivial finite group $G$, denoted $\sigma(G)$, is the smallest number of proper subgroups of $G$ whose set-theoretic union equals $G$. In this article, we focus on a dual…

### On the minimal dimension of a finite simple group

- MathematicsJ. Comb. Theory, Ser. A
- 2020

### Irredundant generating sets and dimension-like invariants of the finite group

- Mathematics
- 2017

Whiston proved that the maximum size of an irredundant generating set in the symmetric group $S_n$ is $n-1$, and Cameron and Cara characterized all irredundant generating sets of $S_n$ that achieve…

### Maximal irredundant families of minimal size in the alternating group

- MathematicsArchiv der Mathematik
- 2019

Let G be a finite group. A family $${\mathcal {M}}$$M of maximal subgroups of G is called “irredundant” if its intersection is not equal to the intersection of any proper subfamily. $${\mathcal…

### Maximal subgroups of finite soluble groups in general position

- Mathematics
- 2015

For a finite group G we investigate the difference between the maximum size $${{\mathrm{MaxDim}}}(G)$$MaxDim(G) of an “independent” family of maximal subgroups of G and maximum size m(G) of an…

### The Replacement Property of PSL$(2,p)$ and PSL$(2,p^2)$

- Mathematics
- 2017

In 2014, Benjamin Nachman showed that when $p\equiv$1 mod 8, the 2-dimensional projective linear group over the field of $p$ elements fails the replacement property if the maximal length $m$ of an…

### Maximal subgroups of finite soluble groups in general position

- Materials ScienceAnnali di Matematica Pura ed Applicata (1923 -)
- 2015

For a finite group G we investigate the difference between the maximum size MaxDim(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}…

### Maximal irredundant families of minimal size in the alternating group

- Materials ScienceArchiv der Mathematik
- 2019

Let G be a finite group. A family M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs}…

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