✦ LIBER ✦

Class-Preserving Automorphisms of Finite Groups

✍ Scribed by Martin Hertweck

Publisher: Elsevier Science
Year: 2001
Tongue: English
Weight: 200 KB
Volume: 241
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.2001.8760

No coin nor oath required. For personal study only.

✦ Synopsis

We construct a family F F of Frobenius groups having abelian Sylow subgroups Ž and non-inner, class-preserving automorphisms. We show that any A-group that is, . a finite solvable group with abelian Sylow subgroups has a sub-quotient belonging to F F provided it has a non-inner, class-preserving automorphism. As a consequence, we obtain that for metabelian A-groups, or A-groups with elementary abelian Sylow subgroups, class-preserving automorphisms are necessarily inner automorphisms. The same is true for a finite group whose Sylow subgroups of odd order are all cyclic, and whose Sylow 2-subgroups are either cyclic, dihedral, or generalized quaternion. Some applications are given.

📜 SIMILAR VOLUMES

A note on groups of colour preserving au

A note on groups of colour preserving automorphisms

✍ Ulrike Baumann; Ute Holthaus 📂 Article 📅 2011 🏛 John Wiley and Sons 🌐 English ⚖ 233 KB

## Abstract A perfect edge colouring of a graph is defined by the property that all colour matchings are perfect matchings. Every edge‐coloured graph determines a group of graph automorphisms which preserve the colours of the edges. If the graph is connected, then this group of colour preserving au

Automorphism Subgroups of Finite Index i

Automorphism Subgroups of Finite Index in Algebraic Mapping Class Groups

✍ Warren Dicks; Edward Formanek 📂 Article 📅 1997 🏛 Elsevier Science 🌐 English ⚖ 316 KB

We give an algebraic proof of the Birman᎐Bers theoremᎏan algebraic result whose previous proofs used topology or analysis, and which says that a certain Ž . subgroup of finite index in the algebraic mapping class group of an oriented punctured surface is isomorphic to a certain group of automorphism

On Groups of Automorphisms of Residually

On Groups of Automorphisms of Residually Finite Groups

✍ Ulderico Dardano; Bettina Eick; Martin Menth 📂 Article 📅 2000 🏛 Elsevier Science 🌐 English ⚖ 111 KB

We show that certain properties of groups of automorphisms can be read off from the actions they induce on the finite characteristic quotients of their underlying group G. In particular, we obtain criteria for groups of automorphisms of a Ž . residually finite and soluble minimax -by-finite group G

Endomorphisms of Free Groups That Preser

Endomorphisms of Free Groups That Preserve Automorphic Orbits

✍ Donghi Lee 📂 Article 📅 2002 🏛 Elsevier Science 🌐 English ⚖ 86 KB

We prove that if an endomorphism φ of a free group F n of a finite rank n preserves an automorphic orbit Orb AutF n W with W = 1, i.e., if φ Orb AutF n W ⊆ Orb AutF n W , then φ is an automorphism.  2002 Elsevier Science (USA)

On Almost Regular Automorphisms of Finit

On Almost Regular Automorphisms of Finite p-Groups

✍ A. Jaikin-Zapirain 📂 Article 📅 2000 🏛 Elsevier Science 🌐 English ⚖ 143 KB

In this paper we prove that there are functions f ( p, m, n) and h(m) such that any finite p-group with an automorphism of order p n , whose centralizer has p m points, has a subgroup of derived length h(m) and index f ( p, m, n). This result gives a positive answer to a problem raised by E. I. Khuk

Noninner Automorphisms of Order p of Fin

Noninner Automorphisms of Order p of Finite p-Groups

✍ Marian Deaconescu; Gheorghe Silberberg 📂 Article 📅 2002 🏛 Elsevier Science 🌐 English ⚖ 64 KB

The main result of this paper shows that if G is a finite nonabelian p-group and if C G Z Φ G = Φ G , then G has a noninner automorphism of order p which fixes Φ G . This reduces the verification of the longstanding conjecture that every finite nonabelian p-group G has a noninner automorphism of ord