Let G be a finite group and let c(G) be the number of cyclic subgroups of G. We study the function α(G) = c(G) / | G|. We explore its basic properties and we point out a connection with the probability of commutation. For many families F of groups we characterize the groups G∈ F for which α(G) is maximal and we classify the groups G for which α(G) > 3 / 4. We also study the number of cyclic subgroups of a direct power of a given group deducing an asymptotic result and we characterize the equality α(G) = α(G/ N) when G / N is a symmetric group.
On the Number of Cyclic Subgroups of a Finite Group
Garonzi, Martino
;
2018
Abstract
Let G be a finite group and let c(G) be the number of cyclic subgroups of G. We study the function α(G) = c(G) / | G|. We explore its basic properties and we point out a connection with the probability of commutation. For many families F of groups we characterize the groups G∈ F for which α(G) is maximal and we classify the groups G for which α(G) > 3 / 4. We also study the number of cyclic subgroups of a direct power of a given group deducing an asymptotic result and we characterize the equality α(G) = α(G/ N) when G / N is a symmetric group.File in questo prodotto:
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