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A. Kulakoff (Math. Ann.  104  (1931), 778-793) proved that for      p &gt; 2 p &gt; 2       the number of solutions of the equation      x p k = e {x^{{p^k}}} = e       (     e e       is a unit element of      G G      ) in a finite noncyclic      p p      -group      G G       is divisible by      p k + 1 {p^{k + 1}}       if      exp ⁡  G ≥  p k \operatorname {exp} G \geq {p^k}      . In this note we consider the number      N ( a , G , k ) N(a,G,k)       of solutions of the equation      x p k = a {x^{{p^k}}} = a       in      G , a ∈  G G,\;a \in G      . Our results cover the case      p = 2 p = 2       also.

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