We prove that any maximal group in the free burnside semigroup defined by the equation x sup n=x sup n+m for any n large than or equal to 1 and any m large than equal to 1 is a free Burnside group satisfying x sup 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 -class containing the group. For n=2 and for every m large than or equal to 2 we present examples with 2m-1 generators. Hence, ihn these cases, we have infinite maximal groups for large enough m. This allows us to prove important properties of Burnise semigroups for the case n=2, which was almost completely unknown until now.Surprisingly, the case n=2 presents simutaneously the complexities ofthe cases n=1 and n large than or equal to 3: the maximal groups are cyclic of order m for n large than or equla to 3 but they can have more generators andbe infinte for n less than or equal to 2; theere are exactly classes and they are easily characterized for n=1 but there are infinitely many -classes and they are difficult to characterize for n large than or equal to 2.
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机译：我们证明，对于任何大于或等于1的n和大于等于1的任何m，由方程x sup n = x sup n + m定义的自由Burnside半群中的任何最大群都是满足x sup m的自由Burnside群= 1。我们表明，在一组描述良好的生成器中，该组是自由的，其发电机的基数是与包含该组的-class相关的图的圈数。对于n = 2，并且对于每m个大于或等于2的m，我们给出具有2m-1个生成器的示例。因此，在这些情况下，对于足够大的m，我们有无限的最大组。这使我们能够证明对于情况n = 2直到现在几乎完全未知的Burnise半群的重要性质。令人惊讶的是，情况n = 2同时呈现了情况n = 1和n大于或等于3的复杂度：当n大于或等于3时，最大基团是m阶的循环，但是它们可以具有更多的生成器，并且对于n小于或等于2是无限的。 theere恰好是类，很容易针对n = 1进行表征，但是存在无限多个-class，并且对于n大于或等于2的n很难进行表征。
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