Abstract
A. Kulakoff [9] proved that forp>2 the numberN k =N k (G) of solutions of the equationx p k=e in a non-cyclicp-groupG is divisible byp k+1. This result is a generalization of the well-known theorem of G. A. Miller asserting that the numberC k =C k (G) of cyclic subgroups of orderp k>p>2 is divisible byp. In this note we show that, as a rule: (1) ifk>1, thenN k ≡0(modp k+p); (2) ifk>2, thenC k ≡0(modp p). These facts are generalizations of many results from [1–5,8,9].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ya. G. Berkovich,On the number of solutions of the equation x p k=a in a finite p-group, to appear.
Ya. G. Berkovich,A generalization of the theorems of Hall and Blackburn and their applications to nonregular p-groups, Math. USSR Izvestiya5 (1971), 815–844.
Ya. G. Berkovich,On p-groups of finite order, Siberian Math. J.9 (1968), 963–978.
Ya. G. Berkovich,Finite p-groups containing at most p p − 1Cyclic subgroups of order p n, Voprosy teorii grup i gomologicheskoi algebry2 (1979), Yaroslavl.
Ya. G. Berkovich, II, Matematicheski analis i ego prilozenia, Rostov University (1981), 10–16, Rostov-Don (in Russian).
N. Blackburn,On a special class of p-groups, Acta Math.100 (1958), 45–92.
N. Blackburn,Generalizations of certain elementary theorems on p-groups, Proc. London Math. Soc.11 (1961), 1–22.
N. Blackburn,Note on a paper of Berkovich, J. Algebra24 (1973), 323–334.
A. Kulakoff,Uber die Anzahl der eigentlichen Untergruppen und der Elemente von gegeberen Ordnung in p-Gruppen, Math. Ann.104 (1931), 778–793.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Berkovich, Y.G. On the number of elements of given order in a finitep-group. Israel J. Math. 73, 107–112 (1991). https://doi.org/10.1007/BF02773429
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02773429