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Probability of cyclicity of chains in finite groups. (English) Zbl 1538.20074

Summary: We introduce the probability of cyclicity of chains in finite groups and among other results we give explicit formulas for such probability of dihedral groups \(D_{2n}\), quasi-dihedral groups \(QD_{2n}\), generalized quaternion groups \(Q_{2n}\), and modular \(p\)-groups \(M_{p^n}\).

MSC:

20P05 Probabilistic methods in group theory
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20F05 Generators, relations, and presentations of groups
20D30 Series and lattices of subgroups