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On the number of solutions of the equation $ x\sp {p\sp k}=a$ in a finite $ p$-group


Author: Yakov G. Berkovich
Journal: Proc. Amer. Math. Soc. 116 (1992), 585-590
MSC: Primary 20D60; Secondary 20D15
DOI: https://doi.org/10.1090/S0002-9939-1992-1093592-9
MathSciNet review: 1093592
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Abstract: A. Kulakoff (Math. Ann. 104 (1931), 778-793) proved that for $ p > 2$ the number of solutions of the equation $ {x^{{p^k}}} = e$ ($ e$ is a unit element of $ G$) in a finite noncyclic $ p$-group $ G$ is divisible by $ {p^{k + 1}}$ if $ \operatorname{exp} G \geq {p^k}$. In this note we consider the number $ N(a,G,k)$ of solutions of the equation $ {x^{{p^k}}} = a$ in $ G,\;a \in G$. Our results cover the case $ p = 2$ also.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1093592-9
Article copyright: © Copyright 1992 American Mathematical Society

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