Abstract

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p ≧ 0. We study J.-P. Serre's notion of G-complete reducibility for subgroups of G. Specifically, for a subgroup H and a normal subgroup N of H, we look at the relationship between G-complete reducibility of N and of H, and show that these properties are equivalent if H/N is linearly reductive, generalizing a result of Serre. We also study the case when H = MN with M a G-completely reducible subgroup of G which normalizes N. In our principal result we show that if G is connected, N and M are connected commuting G-completely reducible subgroups of G, and p is good for G, then H = MN is also G-completely reducible.