  In the present paper, we prove the first part in the standard description of
groups $H$ lying between $m$-th exterior power of elementary group $E(n,R)$ and
the general linear group $GL_{\binom{n}{m}}(R)$. We study structure of the
exterior power of elementary group and its relative analog
$E\left(\binom{n}{m},R,A\right)$. In the considering case $n \geq 3m$, the
description is explained by the classical notion of level: for every such $H$
we find unique ideal $A$ of the ring $R$. Motivated by the problem, we prove
the coincidence of the following groups: normalizer of the exterior power of
elementary group, normalizer of the exterior power of special linear group,
transporter of the exterior power of elementary group into the exterior power
of special linear group, and an exterior power of general linear group. This
result mainly follows from the found explicit equations for the exterior power
of algebraic group scheme $GL_n(\_)$.
