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 Zi + 1/Zi is the center of G/Zi, 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 Zλ(G) = G, but Zα(G) ≠ G for α < λ, (Gluškov 1952) and (McLain 1956).
Read more about this topic: Central Series
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