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Finite non-cyclic \(p\)-groups whose number of subgroups is minimal. (English) Zbl 1445.20014

Let \(p\) be a prime. It is clear that among all finite \(p\)-groups of a given order \(p^n\), the elementary abelian group \(C_p^n\) has the most subgroups. H. Qu [Isr. J. Math. 195, Part B, 773–781 (2013; Zbl 1285.20017), Theorem 1.4] and M. Tărnăuceanu [J. Group Theory 22, No. 3, 505–514 (2019; Zbl 1441.20012), Theorem 1.1] identified the unique \(p\)-group of order \(p^n\) with the second-most subgroups: It is of the form \(G_p\times C_p^{n-3}\) where \(G_2:=\operatorname{D}_8\) and for odd \(p\), \(G_p\) is the unique group of order \(p^3\) and exponent \(p\).
On the other hand, it is clear that the cyclic group \(C_{p^n}\) has the smallest number of subgroups among all groups of order \(p^n\). In this paper, the authors find those groups of order \(p^n\) that have the second-smallest number of subgroups, thus providing analogues of Qu’s [loc. cit.] and Tărnăuceanu’s [loc. cit.] results “at the opposite end of the spectrum”. They show that unless \(p=2\) and \(n\leq 4\), there are exactly two groups of order \(p^n\) with the second-least subgroups among all groups of order \(p^n\), namely \(C_{p^{n-1}}\times C_p\) and \(M_{p^n}:=\langle x,y\mid x^{p^{n-1}}=y^p=1,x^y=x^{1+p^{n-2}}\rangle\). The proof proceeds roughly in the following steps:
(1) Using A. Kulakoff’s result [Math. Ann. 104, 778–793 (1931; Zbl 0001.38602), Satz 1] for \(p>2\) and Frobenius’ generalization of Sylow’s theorems paired with [Y. Berkovich, Groups of prime power order. Vol. 1. Berlin: Walter de Gruyter (2008; Zbl 1168.20001), Proposition 1.3] for \(p=2\), one finds that a group \(G\) of order \(p^n\) with the second-least subgroups must have exactly \(p+1\) subgroups of order \(p^k\) for each \(k\in\{1,\ldots,n-1\}\).
(2) In particular, since \(G\) has at least one normal subgroup of each given order, it follows that all non-normal subgroups of \(G\) with the same order are conjugate. Such groups are classified in a theorem of Y. Berkovich and Z. Janko [Groups of prime power order. Vol. 2. Berlin: Walter de Gruyter (2008; Zbl 1168.20002), Theorem 58.3].
(3) Excluding the nonabelian Dedekind groups for \(p=2\) and using a well-known formula for the number of subgroups of a given type in the abelian case, [L. M. Butler, Proc. Am. Math. Soc. 101, 771–775 (1987; Zbl 0647.20053), Equation (1)], one obtains that \(G\) must indeed be isomorphic to one of \(C_{p^{n-1}}\times C_p\) or \(M_{p^n}\).
(4) For the reverse implication, one can invoke a result of W. Lindenberg [J. Reine Angew. Math. 241, 118–146 (1970; Zbl 0194.03703), Folgerung 3.4].

MSC:

20D15 Finite nilpotent groups, \(p\)-groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups

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References:

[1] Berkovich, Y.: Groups of prime-power order. Vol. 1, volume 46 of de Gruyter Expositions in Mathematics. Walter de Gruyter GmbH & Co. KG, Berlin (2008). [With a foreword by Zvonimir Janko] · Zbl 1168.20001
[2] Berkovich, Y., Janko, Z.: Groups of prime-power order. Vol. 2, volume 47 of de Gruyter Expositions in Mathematics. Walter de Gruyter GmbH & Co. KG, Berlin (2008) · Zbl 1168.20002
[3] Burnside, W., On some properties of groups whose orders are powers of primes, Proc. Lond. Math. Soc., 2, 11, 225-245 (1913) · JFM 43.0198.02 · doi:10.1112/plms/s2-11.1.225
[4] Butler, Lm, A unimodality result in the enumeration of subgroups of a finite abelian group, Proc. Am. Math. Soc., 101, 4, 771-775 (1987) · Zbl 0647.20053 · doi:10.1090/S0002-9939-1987-0911049-8
[5] Calhoun, Wc, Counting the subgroups of some finite groups, Am. Math. Mon., 94, 1, 54-59 (1987) · Zbl 1303.20024 · doi:10.1080/00029890.1987.12000593
[6] Kulakoff, A., Über die Anzahl der eigentlichen Untergruppen und der Elemente von gegebener Ordnung in \(p\)-Gruppen, Math. Ann., 104, 1, 778-793 (1931) · Zbl 0001.38602 · doi:10.1007/BF01457969
[7] Lindenberg, W., Über die Struktur zerfallender bizyklischer \(p\)-Gruppen, J. Reine Angew. Math., 241, 118-146 (1970) · Zbl 0194.03703
[8] Qu, H., Finite non-elementary abelian \(p\)-groups whose number of subgroups is maximal, Israel J. Math., 195, 2, 773-781 (2013) · Zbl 1285.20017 · doi:10.1007/s11856-012-0114-0
[9] Tărnăuceanu, M., On a conjecture by Haipeng Qu, J. Group Theory, 22, 3, 505-514 (2019) · Zbl 1441.20012 · doi:10.1515/jgth-2018-0202
[10] The GAP Group. GAP-Groups, Algorithms, and Programming, Version 4.8.7 (2017)
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