# Hypercentre

From Encyclopedia of Mathematics

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.

#### Comments

#### References

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

**How to Cite This Entry:**

Hypercentre.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Hypercentre&oldid=36895

This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article