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Izv. Akad. Nauk SSSR Ser. Mat., 1971, Volume 35, Issue 4, Pages 800–830 (Mi izv2058)  

This article is cited in 11 scientific papers (total in 11 papers)

A generalization of the theorems of Hall and Blackburn and their applications to nonregular $p$-groups

Ya. G. Berkovich


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 $x^p=1$ in $G$ is equal to $p^p + k(p-1)p^p$, where $k$ is a nonnegative integer; 2) if $n>1$, then the number of solutions of the equation $x^{p^n}=1$ in $G$ is divisible by $p^{n+p-1}$. 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.

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English version:
Mathematics of the USSR-Izvestiya, 1971, 5:4, 815–844

Bibliographic databases:

UDC: 519.44
MSC: Primary 20D15; Secondary 20D25
Received: 09.03.1970

Citation: Ya. G. Berkovich, “A generalization of the theorems of Hall and Blackburn and their applications to nonregular $p$-groups”, Izv. Akad. Nauk SSSR Ser. Mat., 35:4 (1971), 800–830; Math. USSR-Izv., 5:4 (1971), 815–844

Citation in format AMSBIB
\Bibitem{Ber71}
\by Ya.~G.~Berkovich
\paper A~generalization of the theorems of Hall and Blackburn and their applications to nonregular $p$-groups
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1971
\vol 35
\issue 4
\pages 800--830
\mathnet{http://mi.mathnet.ru/izv2058}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=294495}
\zmath{https://zbmath.org/?q=an:0257.20014}
\transl
\jour Math. USSR-Izv.
\yr 1971
\vol 5
\issue 4
\pages 815--844
\crossref{https://doi.org/10.1070/IM1971v005n04ABEH001118}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Joseph A Gallian, “On the Hughes conjecture”, Journal of Algebra, 34:1 (1975), 54  crossref
    2. W. Mack Hill, “Frattini subgroups and supernilpotent groups”, Isr J Math, 26:1 (1977), 68  crossref  mathscinet  zmath
    3. G.L.. Lange, “Two-generator Frattini subgroups of finitep-groups”, Israel J. Math, 29:4 (1978), 357  crossref
    4. H. Bechtell, “On nonnilpotent inseparable groups of order pnqm”, Journal of Algebra, 75:1 (1982), 223  crossref
    5. Yakov G. Berkovich, “On the number of elements of given order in a finitep-group”, Isr J Math, 73:1 (1991), 107  crossref  mathscinet  zmath  isi
    6. Avinoam Mann, “On p-groups whose maximal subgroups are isomorphic”, J Austral Math Soc, 59:2 (1995), 143  crossref
    7. Yakov Berkovich, “On Abelian Subgroups ofp-Groups”, Journal of Algebra, 199:1 (1998), 262  crossref
    8. I. A. Sagirov, “Degrees of irreducible characters of the Suzuki 2-groups”, Math. Notes, 66:2 (1999), 203–207  mathnet  crossref  crossref  mathscinet  zmath  isi
    9. Yakov Berkovich, “On Subgroups of Finite p-Groups”, Journal of Algebra, 224:2 (2000), 198  crossref
    10. Yakov Berkovich, “On Subgroups and Epimorphic Images of Finite p-Groups”, Journal of Algebra, 248:2 (2002), 472  crossref
    11. Yakov Berkovich, “Finite p-groups in which some subgroups are generated by elements of order p”, Glas Mat Ser III, 44:1 (2009), 167  crossref
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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