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June 19, 2026Bulletin of the Australian Mathematical Society0 citationsOpen Access

Characterising Finite Solvable Groups Through the Nilpotency Probability

ALAndrea Lucchini

Key Points

  • To determine the conditions under which a finite group is solvable based on the nilpotency probability.
  • Analyzed the nilpotency probability of randomly chosen elements in finite groups.
  • Proved that if the nilpotency probability exceeds 1/12, the group is solvable.
  • Established a threshold for solvability: if ν(G) > 1/12, then G is solvable.

Abstract

Abstract Given a finite group G, we denote by ν (G) (G) nu left parenthesis upper G right parenthesis the probability that two randomly chosen elements of G generate a nilpotent subgroup. We prove that if ν (G) > 1 / 12, (G) > 1/12, nu left parenthesis upper G right parenthesis greater than 1 divided by 12 comma then G is solvable.

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Cite This Study

Andrea Lucchini (2026) studied this question.

synapsesocial.com/papers/6a34df2365a5b0777af2e449https://doi.org/10.1017/s0004972726101476
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