Overgroups of exterior powers of an elementary group. levels.

We prove a first part of the standard description of groups H lying between an exterior power of an elementary group $ {m}\operatorname {E}_n(R) $ m E n ⁡ (R) and a general linear group $ \operatorname {GL}_{\binom {n}{m}}(R) $ GL (n m) ⁡ (R) for a commutative ring $ R, 2 \in R^{*} $ R , 2 ∈ R ∗ and $ n \geqslant 3m $ n ⩾ 3 m. The description uses the classical notion of a level: for every group H we find a unique ideal A of the ground ring R, which describes H. [ABSTRACT FROM AUTHOR]