Analogues of the fundamental theorem of projective geometry in group theory. I.
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.
scientific article; zbMATH DE number 4008643
scientific article