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Analoga des Fundamentalsatzes der Projektiven Geometrie in der Gruppentheorie. II. (Analogues of the fundamental theorem of projective geometry in group theory. II). (German) Zbl 0636.20016

This is an immediate continuation of part I [ibid. 77, 255-303 (1987; Zbl 0622.20012)]. We summarize theorems 4-6. Let \(G=G_ 1\times...\times G_ r\) be a finite p-group, and let \(\phi\) be an isomorphism of the subgroup lattice of G such that \(G^{\phi}=G_ 1^{\phi}\times...\times G_ r^{\phi}\). Then \(\phi\) is induced by a group isomorphism, if one of the following three conditions holds. (1) \(r=3\), \(G_ 1\cong G_ 2\cong G_ 3\) is metabelian, of maximal class, and has exponent p, and the centralizer \(C_{G_ 1}(G_ 1'/[[G_ 1',G_ 1],G_ 1])\) has class at most two. (2) \(p\geq 5\), \(r\geq p-2\), \(G_ 1\cong G_ 2\cong...\cong G_ r\) is metabelian, but not abelian, and has exponent p. (3) \(p\geq 3\), \(r\geq 3\), \(G_ 1\cong G_ 2\cong...\cong G_ r\) has class at most 2.
Complicated counterexamples show that these conditions cannot be relaxed very much; e.g. the condition on the centralizer in (1) cannot be dropped. In fact, for infinitely many natural numbers m, there exists a metabelian group G of maximal class and exponent p with the following property: every automorphism of the subgroup lattice of \(G^{m+1}\) is induced by a group automorphism, but the corresponding statement for \(G^ m\) is false.
Reviewer: Th.Grundhöfer

MSC:

20D30 Series and lattices of subgroups
20D15 Finite nilpotent groups, \(p\)-groups

Citations:

Zbl 0622.20012
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References:

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