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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
##### Keywords:
finite Abelian groups; random generators; probabilities
##### Citations:
Zbl 0435.20055; Zbl 1274.20110
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