Maximal groups in free Burnside semigroups
We prove that any maximal group in the free Burnside semigroup defined 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 ℑ-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 infinite 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 infinite for n ≤ 2; there are exactly 2¦A¦ ℑ-classes and they are easily characterized for n=1 but there are infinitely many, ℑ-classes and they are difficult to characterize for n ≥ 2.
KeywordsWord Problem Local Group Short Word Congruence Class Finiteness Property
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