On the nilpotent probability and supersolvability of finite groups
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On the nilpotent probability and supersolvability of finite groups Huaquan Wei1 · Huilong Gu1 · Jiao Li1 · Liying Yang2 Received: 21 February 2020 / Accepted: 4 November 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract Let G be a finite group. We denote by N il G (x) the set of elements y ∈ G such that x, y is a nilpotent subgroup and by ν1 (G) and ν(G) the probability that two randomly chosen elements of G respectively generate an abelian subgroup and a nilpotent subgroup. A group G is called an N -group if N il G (x) is a group for all x ∈ G. It is proved that G is supersolvable if there exists a normal subgroup H such that either ν1 (H ) > 13 |G : H | or G is an N -group and ν(H ) > 13 |G : H |. Keywords Finite group · Commuting probability · Nilpotent probability · N -group · Supersolvable group Mathematics Subject Classification 20D10 · 20D20
1 Introduction All groups considered are finite. Let G be a group. We denote by π(G) and N il G (x) the set of prime divisors of |G| and the set of elements y ∈ G such that x, y is a nilpotent subgroup, respectively. By ν1 (G) and ν(G), we denote the probability that two randomly chosen elements of G respectively generate an abelian subgroup and
Communicated by John S. Wilson. Projects supported by NSF of China (12061011), NSF of Guangxi (2019JJD110010) and SRF of Guangxi University (XGZ130761).
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Huaquan Wei [email protected] Liying Yang [email protected]
1
College of Mathematics and Information Sciences, Guangxi University, Nanning 530004, China
2
School of Mathematics and Statistics, Nanning Normal University, Nanning 530299, China
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H. Wei et al.
a nilpotent subgroup. For convenient, we call ν(G) the nilpotent probability of G. Clearly, ν1 (G) ≤ ν(G). Probability theory has played an important role in the study of finite groups. An early example concerns ν1 (G), the commuting probability of G. This concept was introduced by Erd˝os and Turán [1] and it was proved that ν1 (G) = k(G) |G| , where k(G) is the number of conjugacy classes of G. Since then, many scholars have been studied the influence of this probability on the structure of finite groups. For example, Gustafson in [2] showed that ν1 (G) ≤ 58 for any non-abelian group G. Barry et al. in [3] gave ν1 (G) ≤ 13 for any non-supersolvable group G. Guralnick and Robinson 1 for any finite non-abelian simple in [4] stated that Dixon observed that ν1 (G) ≤ 12 group G (this was submitted by Dixon as a problem in Canadian Math. Bulletin, 13 (1970), with his own solution appearing in 1973). In the same paper, they extended this result to non-solvable groups and pointed out many numerical properties of ν1 (G) with explicit bounds. The nilpotent probability ν(G) of finite groups G have been investigated. For instance, Guralnick and Wilson in [5] proved that ν(G) ≤ 21 for any non-nilpotent group G. Fulman et al. in [6] obtained the same result with the additional hypothesis of solvability. Wilson in [7] proved that ν(G) → 0 as |G : F(G)| → ∞, where F(G) is the Fitting subgr
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