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.

中文翻译：

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有限群的循环子群数的第二个最小值/最大值

让$C(G)$是有限群的循环子群的偏集$G$然后让$\mathscr{P}$成为一类$p$- 订单组$p^{n}$($n\geq 3$）。考虑函数$\unicode[STIX]{x1D6FC}:\mathscr{P}\longrightarrow (0,1]$由$\unicode[STIX]{x1D6FC}(G)=|C(G)|/|G|$. 在本文中，我们确定第二个最小值$\unicode[STIX]{x1D6FC}$，以及相应的最小点。由于找到第二个最大值的问题$\unicode[STIX]{x1D6FC}$已解决$p=2$，我们在确定第二个最大值时关注奇素数的情况。