Abstract
In §2, we prove that if a 2-group G and all its nonabelian maximal sub-groups are two-generator, then G is either metacyclic or minimal non-abelian. In §3, we consider a similar question for p > 2. In §4 the 2-groups all of whose minimal nonabelian subgroups have order 16 and a cyclic subgroup of index 2, are classified. It is proved, in §5, that if G is a nonmetacyclic two-generator 2-group and A, B, C are all its maximal subgroups with d(A) ≤ d(B) ≤ d(C), then d(C) = 3 and either d(A) = d(B) = 3 (this occurs if and only if G/G′ has no cyclic subgroup of index 2) or else d(A) = d(B) = 2. Some information on the last case is obtained in Theorem 5.3.
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References
Y. Berkovich, On subgroups and epimorphic images of finite p-groups, Journal of Algebra 248 (2002), 472–553.
Y. Berkovich, Short proofs of some basic characterization theorems of finite p-group theory, Glasnik Matematicki 41(61) (2006), 239–258.
Y. Berkovich, On the number of subgroups of given order in a p-group of exponent p, Proceedings of the American Mathematical Society 109, 4 (1990), 875–879.
Y. Berkovich, On abelian subgroups of p-groups, Journal of Algebra 199 (1998), 262–280.
Y. Berkovich and Z. Janko, Structure of finite groups with given subgroups, Contemporary Mathematics 402 (2006), 13–93.
Ya. G. Berkovich and E. M. Zhmud’, Characters of Finite Groups, Part 1, Translations of Mathematical Monographs, vol. 172, American Mathematical Society, Providence, RI, 1998.
N. Blackburn, On a special class of p-groups, Acta Mathematica 100 (1958), 45–92.
N. Blackburn, Generalizations of certain elementary theorems on p-groups, Proceedings of the London Mathematical Society 11 (1961), 1–22.
I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1967.
Z. Janko, On finite nonabelian 2-groups all of whose minimal nonabelian subgroups are of exponent 4, Journal of Algebra 315 (2007), 801–808.
Z. Janko, Finite 2-groups with exactly one nonmetacyclic maximal subgroup, Israel Journal of Mathematics 166 (2008), 313–347.
L. Redei, Das “schiefe Produkt” in der Gruppentheorie mit Anwendung auf die endlichen nichtkommutativen Gruppen mit lauter kommutativen echten Untergruppen und die Ordnungzahlen, zu denen nur kommutative Gruppen Gehören, Commentarii Mathematici Helvetici 20 (1947), 225–267.
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Berkovich, Y., Janko, Z. On subgroups of finite p-groups. Isr. J. Math. 171, 29–49 (2009). https://doi.org/10.1007/s11856-009-0038-5
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DOI: https://doi.org/10.1007/s11856-009-0038-5