The probability of randomly generating finite abelian groups

Carrico, Tyler

doi:10.2140/involve.2013.6.431

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The probability of randomly generating finite abelian groups

Tyler Carrico

Vol. 6 (2013), No. 4, 431–436

DOI: 10.2140/involve.2013.6.431

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 $ℤ_{p^{n_{1}}} \oplus \dots \oplus ℤ_{p^{n_{k}}}$ , 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

Mathematical Subject Classification 2010

Primary: 20P05

Milestones

Received: 26 July 2012

Revised: 26 October 2012

Accepted: 13 November 2012

Published: 8 October 2013

Communicated by Joseph Gallian

Authors

Tyler Carrico
202-0004 Tokyo Nishitokyo-shi Shimohoya 3-11-23 Japan

Vol. 6, No. 4, 2013