## Involve: A Journal of Mathematics

• Involve
• Volume 6, Number 4 (2013), 431-436.

### The probability of randomly generating finite abelian groups

Tyler Carrico

#### 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 $ℤpm⊕ℤpn$, where $m,n∈ℕ$ and  $p$ is prime, is given, and the result is extended to groups of the form $ℤpn1⊕⋯⊕ℤpnk$, where  $ni,k∈ℕ$ 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.

#### Article information

Source
Involve, Volume 6, Number 4 (2013), 431-436.

Dates
Received: 26 July 2012
Revised: 26 October 2012
Accepted: 13 November 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513733609

Digital Object Identifier
doi:10.2140/involve.2013.6.431

Mathematical Reviews number (MathSciNet)
MR3115976

Zentralblatt MATH identifier
1280.20069

Subjects
Primary: 20P05: Probabilistic methods in group theory [See also 60Bxx]

Keywords
abelian group generate probability

#### Citation

Carrico, Tyler. The probability of randomly generating finite abelian groups. Involve 6 (2013), no. 4, 431--436. doi:10.2140/involve.2013.6.431. https://projecteuclid.org/euclid.involve/1513733609

#### References

• J. A. Gallian, Contemporary abstract algebra, 6th ed., Houghton Mifflin, Boston and New York, 2006.
• D. L. Massari, “The probability of generating a cyclic group”, Pi Mu Epsilon Journal 7:1 (1979), 3–6.
• K. L. Patti, “The probability of randomly generating a finite group”, Pi Mu Epsilon Journal 11:6 (2002), 313–316.