Central Series - Upper Central Series

The upper central series (or ascending central series) of a group G is the sequence of subgroups

where each successive group is defined by:

and is called the ith center of G (respectively, second center, third center, etc.). In this case, Z₁ is the center of G, and for each successive group, the factor group Z_{i + 1}/Z_i is the center of G/Z_i, and is called an upper central series quotient.

For infinite groups, one can continue the upper central series to infinite ordinal numbers via transfinite recursion: for a limit ordinal λ, define

The limit of this process (the union of the higher centers) is called the hypercenter of the group.

If the transfinite upper central series stabilizes at the whole group, then the group is called hypercentral. Hypercentral groups enjoy many properties of nilpotent groups, such as the normalizer condition (the normalizer of a proper subgroup properly contains the subgroup), elements of coprime order commute, and periodic hypercentral groups are the direct sum of their Sylow p-subgroups (Schenkman 1975, Ch. VI.3). For every ordinal λ there is a group G with Z_λ(G) = G, but Z_α(G) ≠ G for α < λ, (Gluškov 1952) and (McLain 1956).