#### Vol. 6, No. 4, 2013

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The probability of randomly generating finite abelian groups

### Tyler Carrico

Vol. 6 (2013), No. 4, 431–436
##### Abstract

Extending the work of Deborah L. Massari and Kimberly L. Patti, 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 ${ℤ}_{{p}^{m}}\oplus {ℤ}_{{p}^{n}}$, where $m,n\in ℕ$ and  $p$ is prime, is given, and the result is extended to groups of the form ${ℤ}_{{p}^{{n}_{1}}}\oplus \cdots \oplus {ℤ}_{{p}^{{n}_{k}}}$, where  ${n}_{i},k\in ℕ$ 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.

##### Keywords
abelian, group, generate, probability
Primary: 20P05