Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On CLT and non-CLT groups

Marius Tărnăuceanu

Abstract

In this note, we prove that for every integer $d\geq 2$ which is not a prime power, there exists a finite solvable group $G$ such that $d\mid |G|$, $π(G)=π(d)$ and $G$ has no subgroup of order $d$. We also introduce the CLT-degree of a finite group and answer two questions about it.

On CLT and non-CLT groups

Abstract

In this note, we prove that for every integer which is not a prime power, there exists a finite solvable group such that , and has no subgroup of order . We also introduce the CLT-degree of a finite group and answer two questions about it.
Paper Structure (3 sections, 8 theorems, 16 equations)

This paper contains 3 sections, 8 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1.1
  3. Proof of Theorem 1.2

Key Result

Theorem 1.1

For every integer $d\geq 2$ which is not a prime power, there exists a finite solvable group $G$ such that $d\mid |G|$, $\pi(G)=\pi(d)$ and $G$ has no subgroup of order $d$.

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Example 2.3
  • Lemma 3.1
  • Lemma 3.2
  • proof