Untergruppen der GL2, welche durch homogene lineare Kongruenzen definiert sind

We generalize a result of Rankin 1]: Let R be a p — adic valuation ring or one of its factor rings and let G be GL₂(R) or SL₂(R). Then for M∈ M₂(R), G_M:={s∈G| trace MS=O} is a group iff trace M=0 and det M satisfies a simple condition (det M=O in most cases). We give similar conditions for several homogeneous linear equations defining subgroups of G.