Involve: A Journal of Mathematics

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

The probability of randomly generating finite abelian groups

Tyler Carrico

Full-text: Open access


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 pmpn, where m,n and  p is prime, is given, and the result is extended to groups of the form pn1pnk, where  ni,k 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

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

abelian group generate probability


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

Export citation


  • 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.