Describing Finite Groups by Short First-order Sentences
نویسنده
چکیده
We say that a class of finite structures for a finite signature is r-compressible if each structure G in the class has a first-order description of size at most O(r(|G|)). We show that the class of finite simple groups is log-compressible, and the class of all finite groups is log-compressible. The result relies on the classification of finite simple groups, the bi-interpretability of the small Ree groups with finite difference fields, and the existence of profinite presentations with few relators. We also indicate why the result is close to optimal.
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