Analogues of the fundamental theorem of projective geometry in group theory. I.
type: Item
instance of: scholarly article
title: Analogues of the fundamental theorem of projective geometry in group theory. I.
zbMATH Open document ID: 0622.20012
author: Michael Schenke
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108066
review text: The fundamental theorem of projective geometry says that every isomorphism between subspace lattices of vector spaces is induced by a semilinear bijection. The author establishes analogous theorems for subgroup lattices of some types of finite \(p\)-groups. The main result of part I is the following (Satz 1-3). Let \(p\) be a prime, let \(G=G_ 1\times\cdots\times G_ r\) be a finite group of exponent \(p\) with \(r\geq 2\) and \(| G| \geq p^ 3\), and let \(\varphi\) be an isomorphism of the subgroup lattice of \(G\) which satisfies \(G^{\varphi}=G_ 1^{\varphi}\times\cdots\times G_ r^{\varphi}\) (by 1.3 this assumption may be omitted if some \(G_ i\) is not abelian). Then \(\phi\) is induced by a group isomorphism if at least one the following conditions holds: (1) each \(G_ i\) has nilpotency class 2; (2) \(p=3\) and no \(G_ i\) is abelian; (3) all \(G_ i\) are isomorphic and have nilpotency class at most 4. For the proof it suffices (by Lemma 1.35) to consider the case \(r=2\). Then ideas used by \textit{R. Baer} [Am. J. Math. 61, 1-44 (1939; Zbl 0020.34704)] for the abelian case are extended to non-abelian groups, and this extension is applied to minimal counterexamples to the main result; the details are quite complicated. In addition the author constructs (in 1.38, 1.39 and 1.47) explicit examples of groups which show that conditions (1) and (3) cannot be replaced by weaker assumptions on the nilpotency classes (for \(p>3\)). Part II of the paper (to appear) will contain more results and (counter-) examples of the same type.
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 4008643
zbMATH Keywords: isomorphism of subgroup lattice
zbMATH Keywords: induced by group isomorphism
zbMATH Keywords: subgroup lattices
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: nilpotency class
zbMATH Keywords: minimal counterexamples
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4008643