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In the present paper, we prove the first part in the standard description ofgroups $H$ lying between $m$-th exterior power of elementary group $E(n,R)$ andthe general linear group $GL_{\binom{n}{m}}(R)$. We study structure of theexterior power of elementary group and its relative analog$E\left(\binom{n}{m},R,A\right)$. In the considering case $n \geq 3m$, thedescription 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 provethe coincidence of the following groups: normalizer of the exterior power ofelementary group, normalizer of the exterior power of special linear group,transporter of the exterior power of elementary group into the exterior powerof special linear group, and an exterior power of general linear group. Thisresult mainly follows from the found explicit equations for the exterior powerof algebraic group scheme $GL_n(\_)$.

Overgroups of exterior powers of an elementary group. I. Levels and normalizers