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THE SUBGROUP COMMUTATIVITY DEGREE OF FINITE $P$-GROUPS

Part of: Representation theory of groups Special aspects of infinite or finite groups Probabilistic methods in group theory

Published online by Cambridge University Press:  08 July 2015

MARIUS TĂRNĂUCEANU
MARIUS TĂRNĂUCEANU*
Affiliation:
Faculty of Mathematics, ‘Al. I. Cuza’ University, Iaşi, Romania email tarnauc@uaic.ro
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Abstract

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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.

Keywords

subgroup commutativity degreessubgroup latticesfinite P-groups

MSC classification

Primary: 20D60: Arithmetic and combinatorial problems 20P05: Probabilistic methods in group theory
Secondary: 20D30: Series and lattices of subgroups 20F16: Solvable groups, supersolvable groups 20F18: Nilpotent groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 1 , February 2016 , pp. 37 - 41
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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