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The subgroup commutativity degree of a group $G$ is the probability that two subgroups of $G$ commute, or equivalently that the product of two subgroups is again a subgroup. For the dihedral, quasi-dihedral and generalised quaternion groups (all of 2-power cardinality), the subgroup commutativity degree tends to 0 as the size of the group tends to infinity. This also holds for the family of projective special linear groups over fields of even characteristic and for the family of the simple Suzuki groups. In this short note, we show that the family of finite $P$ -groups also has this property.



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[1]Aivazidis, S., ‘The subgroup permutability degree of projective special linear groups over fields of even characteristic’, J. Group Theory 16 (2013), 383396.
[2]Aivazidis, S., ‘On the subgroup permutability degree of the simple Suzuki groups’, Monatsh. Math. 176 (2015), 335358.
[3]Schmidt, R., Subgroup Lattices of Groups, de Gruyter Expositions in Mathematics, 14 (de Gruyter, Berlin, 1994).
[4]Suzuki, M., Group Theory, I, II (Springer, Berlin, 1982, 1986).
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[7]Tărnăuceanu, M., ‘An arithmetic method of counting the subgroups of a finite abelian group’, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 53(101) (2010), 373386.
[8]Tărnăuceanu, M., ‘Addendum to “Subgroup commutativity degrees of finite groups”’, J. Algebra 337 (2011), 363368.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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