RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Izv. RAN. Ser. Mat.: Year: Volume: Issue: Page: Find

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

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.

Full text: PDF file (3692 kB)
References: PDF file   HTML file

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

Bibliographic databases:

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

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} 

• http://mi.mathnet.ru/eng/izv2058
• http://mi.mathnet.ru/eng/izv/v35/i4/p800

 SHARE:

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
2. W. Mack Hill, “Frattini subgroups and supernilpotent groups”, Isr J Math, 26:1 (1977), 68
3. G.L.. Lange, “Two-generator Frattini subgroups of finitep-groups”, Israel J. Math, 29:4 (1978), 357
4. H. Bechtell, “On nonnilpotent inseparable groups of order pnqm”, Journal of Algebra, 75:1 (1982), 223
5. Yakov G. Berkovich, “On the number of elements of given order in a finitep-group”, Isr J Math, 73:1 (1991), 107
6. Avinoam Mann, “On p-groups whose maximal subgroups are isomorphic”, J Austral Math Soc, 59:2 (1995), 143
7. Yakov Berkovich, “On Abelian Subgroups ofp-Groups”, Journal of Algebra, 199:1 (1998), 262
8. I. A. Sagirov, “Degrees of irreducible characters of the Suzuki 2-groups”, Math. Notes, 66:2 (1999), 203–207
9. Yakov Berkovich, “On Subgroups of Finite p-Groups”, Journal of Algebra, 224:2 (2000), 198
10. Yakov Berkovich, “On Subgroups and Epimorphic Images of Finite p-Groups”, Journal of Algebra, 248:2 (2002), 472
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
•  Number of views: This page: 570 Full text: 77 References: 31 First page: 1