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On some free semigroups, generated by matrices

Piotr Słanina — 2015

Czechoslovak Mathematical Journal

Let A = 1 2 0 1 , B λ = 1 0 λ 1 . We call a complex number λ “semigroup free“ if the semigroup generated by A and B λ is free and “free” if the group generated by A and B λ is free. First families of semigroup free λ ’s were described by J. L. Brenner, A. Charnow (1978). In this paper we enlarge the set of known semigroup free λ ’s. To do it, we use a new version of “Ping-Pong Lemma” for semigroups embeddable in groups. At the end we present most of the known results related to semigroup free and free numbers in a common picture....

Groups Generated by (near) Mutually Engel Periodic Pairs

Piotr SłaninaWitold Tomaszewski — 2007

Bollettino dell'Unione Matematica Italiana

We use notations: [ x , y ] = [ x , 1 y ] and [ x , k + 1 y ] e [ [ x , k y ] , y ] . We consider groups generated by x , y satisfying relations x = [ x , n y ] , y = [ y , n x ] or [ x , y ] = [ x , n y ] , [ y , x ] = [ y , n x ] . We call groups of the first type mep-groups and of the second type nmep-groups. We show many properties and examples of mep- and nmep-groups. We prove that if p is a prime then the group S l 2 ( p ) is a nmep-group. We give the necessary and sufficient conditions for metacyclic group to be a nmep-group and we show that nmep-groups with presentation x , y [ x , y ] = [ x , 2 y ] , [ y , x ] = [ y , 2 x ] , x n , y m are finite.

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