Maximal Groups in Free Burnside Semigroups
نویسنده
چکیده
We prove that any maximal group in the free Burnside semi-group deened by the equation x n = x n+m for any n 1 and any m 1 is a free Burnside group satisfying x m = 1. We show that such group is free over a well described set of generators whose cardinality is the cyclomatic number of a graph associated to the J-class containing the group. For n = 2 and for every m 2 we present examples with 2m ? 1 generators. Hence, in these cases, we have innnite maximal groups for large enough m. This allows us to prove important properties of Burnside semigroups for the case n = 2, which was almost completely unknown until now. Surprisingly, the case n = 2 presents simultaneously the complexities of the cases n = 1 and n 3: the maximal groups are cyclic of order m for n 3 but they can have more generators and be innnite for n 2; there are exactly 2 jAj J-classes and they are easily characterized for n = 1 but there are innnitely many J-classes and they are diicult to characterize for n 2.
منابع مشابه
Geometric Semigroup Theory
Geometric semigroup theory is the systematic investigation of finitely-generated semigroups using the topology and geometry of their associated automata. In this article we show how a number of easily-defined expansions on finite semigroups and automata lead to simplifications of the graphs on which the corresponding finite semigroups act. We show in particular that every finite semigroup can b...
متن کاملSymbolic Dynamics in Free Profinite Semigroups
This is a survey and announcement of recent results on the structure of free profinite semigroups using techniques and results from symbolic dynamics. The intimate connection between uniformly recurrent infinite words and J -maximal regular J -classes is explored to compute the maximal subgroups of the J -classes associated with Arnoux-Rauzy infinite words, which turn out to be free profinite g...
متن کاملOn Finite and Locally Finite Subgroups of Free Burnside Groups of Large Even Exponents
The following basic results on infinite locally finite subgroups of a free m-generŽ . 48 ator Burnside group B m, n of even exponent n, where m ) 1 and n G 2 , n is divisible by 29, are obtained: A clear complete description of all infinite groups that Ž . Ž . are embeddable in B m, n as maximal locally finite subgroups is given. Any Ž . infinite locally finite subgroup L of B m, n is contained...
متن کاملHyperbolic Groups and Completely Simple Semigroups
We begin with a brief introduction to the theory of word hyperbolic groups. We then consider four possible conditions which might reasonably be used as definitions or partial definitions of hyperbolicity in semigroups: having a hyperbolic Cayley graph; having hyperbolic Schützenberger graphs; having a context-free multiplication table; or having word hyperbolic maximal subgroups. Our main resul...
متن کامل