Commutators, generators and conjugacy equations in groups. (English) Zbl 0555.20023
Sei \(G\) eine zweielementig erzeugte Gruppe und \((c,t)\) ein festes, aber ansonsten beliebiges, Erzeugendenpaar von \(G\). Die Autoren diskutieren die folgenden drei Eigenschaften auf ihren inneren Zusammenhang (die in der freien Gruppe vom Rang 2 bekanntermaßen äquivalent sind): (A) Zwei Elemente \(g_ 1,g_ 2\in G\) erzeugen \(G\) genau dann, wenn der Kommutator \([g_ 1,g_ 2]=g_ 1^{-1}g_ 2^{-1}g_ 1g_ 2\) konjugiert ist zu \([c,t]\) oder \([c,t]^{-1}\). (B) Für zwei Elemente \(g_ 1,g_ 2\in G\) ist \([g_ 1,g_ 2]\) genau dann konjugiert zu \([c,t]\) oder \([c,t]^{- 1}\), wenn \((g_ 1,g_ 2)\) Nielsen-äquivalent zu \((c,t)\) ist. (C) Jedes Erzeugendenpaar ist Nielsen-äquivalent zu \((c,t)\). Es ist klar, daß je zwei der Eigenschaften (A), (B), (C) die dritte implizieren.
Reviewer: G.Rosenberger
MSC:
20F05 | Generators, relations, and presentations of groups |
20F12 | Commutator calculus |
20F06 | Cancellation theory of groups; application of van Kampen diagrams |
20E05 | Free nonabelian groups |
20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |
Keywords:
two-generator groups; generating pairs; commutator-generator property; presentations; reduced words; small cancellation groups; Nielsen equivalenceReferences:
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