Nilpotent and Solvable Groups

Abstract

We can further extend the Jordan decomposition to nonabehan groups, but we first need an algebraic formulation of commutator subgroups. Let S be an algebraic matrix group, and consider the map S × S → S sending x, y to xyx⁻¹ y⁻¹. The kernel I ₁ , of the corresponding map k[S] →k[S] ⊗ k[S] consists of the functions vanishing on all commutators in S; that is, the closed set it defines is the closure the commutators. Similarly we have a map S²ⁿ → S sending x₁, y₁, …, x _n y _n , to x₁ y ₁ x₁⁻¹ y₁ ^−1… x _n ⁻¹y _n ⁻¹ and the corresponding map k[S] → ⊗²ⁿk[S] has kernel I _n defining the closure of the products of n commutators. Clearly then I₁ $\supseteq$ I₂ $\supseteq$ I₃ $\supseteq$