Powering map by field characteristic is same in algebra and algebra group

Statement
Suppose $$F$$ is a field of characteristic equal to a prime number $$p$$ and $$N$$ is an associative algebra over $$F$$ that admits an algebra group $$G = \{ 1 + x \mid x \in N \}$$. (This happens, for instance, if $$N$$ is nilpotent). Then, the $$p$$-power map in $$N$$ corresponds to the $$p$$-power map in $$G$$, i.e.:

$$(1 + x)^p = 1 + x^p \ \forall \ x \in N$$

Proof
The proof simply involves expanding using the binomial theorem and then using characteristic $$p$$. Note that we crucially use the fact that 1 commutes with $$x$$ to apply the binomial theorem.