A property of group laws

Abstract

For a word in $n$ letters, in [1] the author introduced a notion: \emph{its standard exponent} and proved that the variety of residually finite groups defined by a word is almost nilpotent if and only if the standard exponent of this word is 1. In this paper we obtain the following result: let $\omega(x_1, \cdots, x_n)$ denote a word in $x_1, \cdots, x_n$. Then both $\omega(x_1, \cdots, x_n)$ and $\omega(x^{m_1}_1, \cdots, x^{m_n}_n)$, where $m_i$ are natural numbers, have the same standard exponents.