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Typical automorphism groups of finite nonrigid structures. (English) Zbl 1373.03050

Summary: We work with a finite relational vocabulary with at least one relation symbol with arity at least 2. Fix any integer \(m > 1\). For almost all finite structures (labelled or unlabelled) such that at least \(m\) elements are moved by some automorphisms, the automorphism group is \((\mathbb{Z}_2)^i\) for some \(i \leq (m+1)/2\); and if some relation symbol has arity at least 3, then the automorphism group is almost always \(\mathbb{Z}_2\).

MSC:

03C13 Model theory of finite structures
60C05 Combinatorial probability
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
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