How Commutative Are Direct Products of Dihedral Groups?

IfGis a finite group, then Pr(G) is the probability that two randomly selected elements ofGcommute. SoGis abelian iff Pr(G) = 1. For any positive integerm, we show that there is a groupGwhich is a direct product of dihedral groups such that Pr(G) = 1/m. We also show that there is a dihedral groupGsu...

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Bibliographic Details
Published inMathematics magazine Vol. 84; no. 2; pp. 137 - 140
Main Authors Clifton, Cody, Guichard, David, Keef, Patrick
Format Magazine Article
LanguageEnglish
Published Mathematical Association of America 01.04.2011
Subjects
Commuting
Direct products
Fractions
Integers
Mathematical theorems
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Summary:IfGis a finite group, then Pr(G) is the probability that two randomly selected elements ofGcommute. SoGis abelian iff Pr(G) = 1. For any positive integerm, we show that there is a groupGwhich is a direct product of dihedral groups such that Pr(G) = 1/m. We also show that there is a dihedral groupGsuch that Pr(G) =m/m′, wherem′ is relatively prime tom.
ISSN:0025-570X
1930-0980
DOI:10.4169/math.mag.84.2.137