Central series

In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, this is an explicit expression that the group is a nilpotent group, and for matrix rings, this is an explicit expression that in some basis the matrix ring consists entirely of upper triangular matrices with constant diagonal.

This article uses the language of group theory; analogous terms are used for Lie algebras.

The lower central series and upper central series (also called the descending central series and ascending central series, respectively), are characteristic series, which, despite the names, are central series if and only if a group is nilpotent.

A central series is a sequence of subgroups

such that the successive quotients are central; that is, [G, A_{i + 1}] ≤ A_i, where [G, H] denotes the commutator subgroup generated by all g⁻¹h⁻¹gh for g in G and h in H. As [G, A_{i + 1}] ≤ A_i ≤ A_{i + 1}, in particular A_{i + 1} is normal in G for each i, and so equivalently we can rephrase the 'central' condition above as: A_{i + 1}/A_i commutes with all of G/A_i.

A central series is analogous in Lie theory to a flag that is strictly preserved by the adjoint action (more prosaically, a basis in which each element is represented by a strictly upper triangular matrix); compare Engel's theorem.

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