Example

Let G = 1 + M_n(m), which is an open subset of M_n(K). If x ∈ G, then det x ∈ 1 + m and hence G ⊂ GL(n, K). Clearly GG ⊂ G. If x = 1 + y with y ∈ M_n(m), the calculation of the inverse of a matrix proves first that x^{-1} ∈ M_n(A); if we write x^{-1} = 1 + y', then y + y' + yy' = 0, hence y' ∈ M_n(m) and therefore x^{-1} ∈ G. Thus G is an open subgroup of GL(n, K). We identify G with m^{n^2} by means of the mapping (\delta_{ij} + y_{ij}) ↦ (y_{ij}). Clearly G is a standard group.