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The probability of randomly generating finite Abelian groups. (English) Zbl 1280.20069
Summary: Extending the work of D. L. Massari [Pi Mu Epsilon J. 7, 3-6 (1979; Zbl 0435.20055)] and K. L. Patti [ibid. 11, No. 6, 313-316 (2002; Zbl 1274.20110)], this paper makes progress toward finding the probability of \(k\) elements randomly chosen without repetition generating a finite Abelian group, where \(k\) is the minimum number of elements required to generate the group. A proof of the formula for finding such probabilities of groups of the form \(\mathbb Z_{p^m}\oplus\mathbb Z_{p^n}\), where \(m,n\in\mathbb N\) and \(p\) is prime, is given, and the result is extended to groups of the form \(\mathbb Z_{p^{n_1}}\oplus\cdots\mathbb Z_{p^{n_k}}\), where \(n_i,k\in\mathbb N\) and \(p\) is prime. Examples demonstrating applications of these formulas are given, and aspects of further generalization to finding the probabilities of randomly generating any finite Abelian group are investigated.
MSC:
20P05 Probabilistic methods in group theory
20K01 Finite abelian groups
20F05 Generators, relations, and presentations of groups
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