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The second minimum/maximum value of the number of cyclic subgroups of finite \(p\)-groups. (English) Zbl 07302540
Summary: 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 \(\alpha : \mathscr{P} \longrightarrow (0, 1]\) given by \(\alpha (G) = |C(G)| / |G|\). In this paper, we determine the second minimum value of \(\alpha\), as well as the corresponding minimum points. Since the problem of finding the second maximum value of \(\alpha\) has been solved for \(p = 2\), we focus on the case of odd primes in determining the second maximum.
MSC:
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D15 Finite nilpotent groups, \(p\)-groups
20D25 Special subgroups (Frattini, Fitting, etc.)
Software:
GAP
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References:
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