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THE SECOND MINIMUM/MAXIMUM VALUE OF THE NUMBER OF CYCLIC SUBGROUPS OF FINITE
$p$-GROUPS
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society / Volume 103 / Issue 1 / February 2021
- Published online by Cambridge University Press:
- 20 April 2020, pp. 96-103
- Print publication:
- February 2021
-
- Article
- Export citation
-
Let
$C(G)$ be the poset of cyclic subgroups of a finite group 
$G$ and let 
$\mathscr{P}$ be the class of 
$p$-groups of order 
$p^{n}$ (
$n\geq 3$). Consider the function 
$\unicode[STIX]{x1D6FC}:\mathscr{P}\longrightarrow (0,1]$ given by 
$\unicode[STIX]{x1D6FC}(G)=|C(G)|/|G|$. In this paper, we determine the second minimum value of 
$\unicode[STIX]{x1D6FC}$, as well as the corresponding minimum points. Since the problem of finding the second maximum value of 
$\unicode[STIX]{x1D6FC}$ has been solved for 
$p=2$, we focus on the case of odd primes in determining the second maximum.
