Automorphism Orbits and Element Orders in Finite Groups : Almost-Solubility and the Monster / Alexander Bors, Michael Giudici, and Cheryl E. Praeger.

Format:

Book

Author/Creator:

Bors, Alexander, author.

Giudici, Michael, 1976- author.

Praeger, Cheryl E., 1948- author.

Series:

Memoirs of the American Mathematical Society ; Volume 287.

Memoirs of the American Mathematical Society Series ; Volume 287

Language:

English

Subjects (All):

Finite groups.

Automorphisms.

Physical Description:

1 online resource (108 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"For a finite group G, we denote by [omega](G) the number of Aut(G)-orbits on G, and by o(G) the number of distinct element orders in G. In this paper, we are primarily concerned with the two quantities d(G) :[equals] [omega](G) - o(G) and q(G) :[equals] [omega](G)/ o(G), each of which may be viewed as a measure for how far G is from being an AT-group in the sense of Zhang (that is, a group with [omega](G) [equals] o(G)). We show that the index [absolute value]G : Rad(G) of the soluble radical Rad(G) of G can be bounded from above both by a function in d(G) and by a function in q(G) and o(Rad(G)). We also obtain a curious quantitative characterization of the Fischer-Griess Monster group M"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. Statement of our main results

1.2. Overview of the proofs of Theorems 1.1.2 and 1.1.3

1.3. Some related open questions

Chapter 2. Notation

Chapter 3. Proof of Theorem 1.1.3

3.1. Sporadic groups

3.2. Alternating groups

3.3. Groups of Lie type

Chapter 4. Proof of Theorem 1.1.2(1)

Chapter 5. Proof of Theorem 1.1.2(2)

5.1. Reduction to semisimple groups

5.2. Two lemmas for working with partitions

5.3. Some auxiliary results on finite simple groups

5.4. Gaining some control over socle cosets in finite semisimple groups

5.5. Another equivalent reformulation of Theorem 1.1.2(2)

5.6. A bit of elementary number theory

5.7. Some results concerning the classes \Hcal_{ ̂, ̂, ̂}

5.8. More restrictions on finite semisimple groups with bounded \q-value

5.9. Completing the proof of Theorem 1.1.2(2)

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Description based on print version record.

Includes bibliographical references.

Other Format:

Print version: Bors, Alexander Automorphism Orbits and Element Orders in Finite Groups: Almost-Solubility and the Monster

ISBN:

9781470475437

147047543X