Geodesic
On the number of solutions of equation \( x^{{p}^{ k}} = 1 \) in a finite group
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 6 (1995) no. 1, pp. 5-12

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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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     title = {On the number of solutions of equation \( x^{{p}^{ k}} = 1 \) in a finite group},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni},
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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. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 6 (1995) no. 1, pp. 5-12. http://geodesic.mathdoc.fr/item/RLIN_1995_9_6_1_a0/