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Proof of a Conjecture by Garrett Birkhoff and Philip Hall on the Automorphisms of a Finite Group

Proof of a Conjecture by Garrett Birkhoff and Philip Hall on the Automorphisms of a Finite Group PROOF OF A CONJECTURE BY GARRETT BIRKHOFF AND PHILIP HALL ON THE AUTOMORPHISMS OF A FINITE GROUP PETER M. NEUMANN 1. The conjecture a a a a a (al) a For a prime powerp define 6(p ) := (p - \){p -p)... (p-p -), so that 9{p ) = \GL(a,p)\ = |Aut(£" a)|, where GL(a,p) is the general linear group of all invertible ax a matrices over the integers modulo p and E a is the elementary abelian group of exponent p and rank a. For a natural number n, if n = \\PV where p ...,p are v k distinct prime numbers, define 6(n) := f ] dip"*). In [1] Garret t Birkhoff and Philip Hall prove that if G is a finite soluble group of order n then |Aut(G)| divides n0(n). They conjecture that this might be true for all finite groups, without any solubility assumptions. Laci Pyber [5] uses a theorem of Aschbacher and Guralnick, a theorem which depends on CFSG, the Classification of the Finite Simple Groups, to prove that |Aut (G)\ ^ n 6(n). I am most grateful to him for sending me a preprint and drawing the Birkhoff-Hall Conjecture to my attention. The http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Proof of a Conjecture by Garrett Birkhoff and Philip Hall on the Automorphisms of a Finite Group

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/27.3.222
Publisher site
See Article on Publisher Site

Abstract

PROOF OF A CONJECTURE BY GARRETT BIRKHOFF AND PHILIP HALL ON THE AUTOMORPHISMS OF A FINITE GROUP PETER M. NEUMANN 1. The conjecture a a a a a (al) a For a prime powerp define 6(p ) := (p - \){p -p)... (p-p -), so that 9{p ) = \GL(a,p)\ = |Aut(£" a)|, where GL(a,p) is the general linear group of all invertible ax a matrices over the integers modulo p and E a is the elementary abelian group of exponent p and rank a. For a natural number n, if n = \\PV where p ...,p are v k distinct prime numbers, define 6(n) := f ] dip"*). In [1] Garret t Birkhoff and Philip Hall prove that if G is a finite soluble group of order n then |Aut(G)| divides n0(n). They conjecture that this might be true for all finite groups, without any solubility assumptions. Laci Pyber [5] uses a theorem of Aschbacher and Guralnick, a theorem which depends on CFSG, the Classification of the Finite Simple Groups, to prove that |Aut (G)\ ^ n 6(n). I am most grateful to him for sending me a preprint and drawing the Birkhoff-Hall Conjecture to my attention. The

Journal

Bulletin of the London Mathematical Society – Wiley

Published: May 1, 1995

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