On the number of solutions of equation $ x^{{p}^{ k}} = 1 $ in a finite group

Yakov Berkovich

On the number of solutions of equation $x^{p^{k}} = 1$ in a finite group

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1995)

Volume: 6, Issue: 1, page 5-12
ISSN: 1120-6330

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Abstract

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Theorem A yields the condition under which the number of solutions of equation

x^{p^{k}} = 1

in a finite

p

-group is divisible by

p^{n + k}

(here

n

is a fixed positive integer). Theorem B which is due to Avinoam Mann generalizes the counting part of the Sylow Theorem. We show in Theorems C and D that congruences for the number of cyclic subgroups of order

p^{k}

which are true for abelian groups hold for more general finite groups (for example for groups with abelian Sylow

p

-subgroups).

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Berkovich, Yakov. "On the number of solutions of equation $ x^{{p}^{ k}} = 1 $ in a finite group." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 6.1 (1995): 5-12. <http://eudml.org/doc/244269>.

@article{Berkovich1995,
abstract = {Theorem A yields the condition under which the number of solutions of equation $ x^\{\{p\}^\{ k\}\} = 1 $ in a finite $ p $-group is divisible by $ p^\{n + k\} $ (here $ n $ is a fixed positive integer). Theorem B which is due to Avinoam Mann generalizes the counting part of the Sylow Theorem. We show in Theorems C and D that congruences for the number of cyclic subgroups of order $ p^\{k\} $ which are true for abelian groups hold for more general finite groups (for example for groups with abelian Sylow $ p $-subgroups).},
author = {Berkovich, Yakov},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Finite groups; p-subgroups; p-elements; number of solutions; congruences; number of cyclic subgroups; finite groups; groups with Abelian Sylow -subgroups},
language = {eng},
month = {3},
number = {1},
pages = {5-12},
publisher = {Accademia Nazionale dei Lincei},
title = {On the number of solutions of equation $ x^\{\{p\}^\{ k\}\} = 1 $ in a finite group},
url = {http://eudml.org/doc/244269},
volume = {6},
year = {1995},
}

TY - JOUR
AU - Berkovich, Yakov
TI - On the number of solutions of equation $ x^{{p}^{ k}} = 1 $ in a finite group
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1995/3//
PB - Accademia Nazionale dei Lincei
VL - 6
IS - 1
SP - 5
EP - 12
AB - Theorem A yields the condition under which the number of solutions of equation $ x^{{p}^{ k}} = 1 $ in a finite $ p $-group is divisible by $ p^{n + k} $ (here $ n $ is a fixed positive integer). Theorem B which is due to Avinoam Mann generalizes the counting part of the Sylow Theorem. We show in Theorems C and D that congruences for the number of cyclic subgroups of order $ p^{k} $ which are true for abelian groups hold for more general finite groups (for example for groups with abelian Sylow $ p $-subgroups).
LA - eng
KW - Finite groups; p-subgroups; p-elements; number of solutions; congruences; number of cyclic subgroups; finite groups; groups with Abelian Sylow -subgroups
UR - http://eudml.org/doc/244269
ER -

References

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BERKOVICH, YA. G., On $p$ -groups of finite order. Sibirsk. Math. J., 9, 6, 1968, 1284-1306 (in Russian). MR241534
BERKOVICH, YA. G., On the number of elements of given order in a finite $p$ -group. Israel J. Math., 73, 1991, 107-112. Zbl0734.20007 MR1119932 DOI10.1007/BF02773429
BERKOVICH, YA. G., Counting theorems for finite $p$ -groups. Arch. Math., 59, 1992, 215-222. Zbl0807.20020 MR1174398 DOI10.1007/BF01197318
BLACKBURN, N., Generalizations of certain elementary theorems on $p$ -groups. Proc. London Math. Soc., 11, 1961, 1-22. Zbl0102.01903 MR122876
DELIGNE, P., Congruences sur le nombre de sous-groupes d'ordre $p^{k}$ dans un groupe fini. Bull. Soc. Math. Belg., 18, 1966, 129-132. Zbl0154.02002 MR202821
HERZOG, M., Counting group elements of order $p$ modulo $p^{2}$ . Proc. Amer. Math. Soc., 66, 1977, 247-250. Zbl0378.20013 MR466316

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