Central Series

A central series is a sequence of subgroups

such that the successive quotients are central; that is, ≤ A_i, where denotes the commutator subgroup generated by all g−1h−1gh for g in G and h in H. As ≤ 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.

A group need not have a central series. In fact, a group has a central series if and only if it is a nilpotent group. If a group has a central series, then there are two central series whose terms are extremal in certain senses. Since A₁ ≤ Z(G), the largest choice for A₁ is precisely A₁ = Z(G). Continuing in this way to choose the largest possible A_{i + 1} given A_i produces what is called the upper central series. Dually, since A_n= G, the commutator subgroup satisfies = ≤ A_{n − 1}. Therefore the minimal choice for A_{n − 1} is . Continuing to choose A_i minimally given A_{i + 1} such that ≤ A_i produces what is called the lower central series. These series can be constructed for any group, and if a group has a central series (is a nilpotent group), these procedures will yield central series.