  Let $ F $ be a finite field and consider $ UT_n $ the algebra of $ n\times n
$ upper triangular matrices over $ F $. In [1], it was proved that every $ G
$-grading is elementary. In [2], the authors classified all nonisomorphic
elementary $ G $-gradings. They also described the set of all $G $-graded
polynomial identities for $ UTn $ when $ F $ is an infinite field. In [3], was
described the all $ G $-graded polynomial identities for $ UT_n $ when $ F $ is
a finite field. In this work, we will discuss the case when $G = \mathbb{Z}_2$,
$ n=2, 3 $ and $ F $ is a finite field.
