The probability of randomly generating finite abelian groups

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 ${\mathbb{Z}}_{p^m}\oplus {\mathbb{Z}}_{p^n}$, where $m,n\in \mathbb{N}$ and $p$ is prime, is given, and the result is extended to groups of the form ${\mathbb{Z}}_{p^{n_1}}\oplus \cdots \oplus {\mathbb{Z}}_{p^{n_k}}$, 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.