  Let $D$ be a division ring with infinite center, $n$ be a positive integer
and $w(x_1,x_2,\cdots, x_m)=1$ be a generalized group identity over the general
linear group $\GL_n(D)$. The aim of this paper is to prove that every subnormal
subgroup of $GL_n(D)$ which satisfies the generalized group identity
$w(x_1,x_2,\cdots, x_m)=1$ is central.
