Hypercentre

A member $Z_\alpha$ of the transfinite upper central series of a group $G$. The first hypercentre $Z_1$ is the centre of the group; suppose that all $Z_\beta$, $\beta < \alpha$, are known, then $Z_\alpha = \cup_{\beta<\alpha} Z_\beta$ if $\alpha$ is a limit ordinal number; $Z_\alpha$ is the complete pre-image of the centre of the quotient group $G/Z_\beta$ if $\alpha = \beta+1$ is a non-limit ordinal number. The hypercentres of a group are locally nilpotent.

References

[a1]	D.J.S. Robinson, "Finiteness condition and generalized soluble groups" , 1–2 , Springer (1972)