## 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.

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