Abstract:
In this work we improve Philip Hall's estimate for the number of cyclic subgroups in a finite p-group. From our result it follows that if a p-group G is not absolutely regular and not a group of maximal class, then 1) the number of solutions of the equation xp=1 in G is equal to pp+k(p−1)pp, where k is a nonnegative integer; 2) if n>1, then the number of solutions of the equation xpn=1 in G is divisible by pn+p−1. This permits us to strengthen important theorems of Hall and Norman Blackburn on the existence of normal subgroups of prime exponent. The latter results in turn permit us to give a factorization of p-groups with absolutely regular Frattini subgroup. Another application is a theorem on the number of subgroups of maximal class in a p-group.
Citation:
Ya. G. Berkovich, “A generalization of the theorems of Hall and Blackburn and their applications to nonregular p-groups”, Math. USSR-Izv., 5:4 (1971), 815–844
\Bibitem{Ber71}
\by Ya.~G.~Berkovich
\paper A~generalization of the theorems of Hall and Blackburn and their applications to nonregular $p$-groups
\jour Math. USSR-Izv.
\yr 1971
\vol 5
\issue 4
\pages 815--844
\mathnet{http://mi.mathnet.ru/eng/im2058}
\crossref{https://doi.org/10.1070/IM1971v005n04ABEH001118}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=294495}
\zmath{https://zbmath.org/?q=an:0257.20014}
Linking options:
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https://doi.org/10.1070/IM1971v005n04ABEH001118
https://www.mathnet.ru/eng/im/v35/i4/p800
This publication is cited in the following 12 articles:
Qinhai Zhang, “Finite p-Groups Whose Subgroups of Given Order Are Isomorphic and Minimal Non-abelian”, Algebra Colloq., 26:01 (2019), 1
Yakov Berkovich, “Finite p-groups in which some subgroups are generated by elements of order p”, Glas Mat Ser III, 44:1 (2009), 167
Yakov Berkovich, “On Subgroups and Epimorphic Images of Finite p-Groups”, Journal of Algebra, 248:2 (2002), 472
Yakov Berkovich, “On Subgroups of Finite p-Groups”, Journal of Algebra, 224:2 (2000), 198
I. A. Sagirov, “Degrees of irreducible characters of the Suzuki 2-groups”, Math. Notes, 66:2 (1999), 203–207