View source for Hypercentre

A member <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048370/h0483701.png" /> of the upper central series of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048370/h0483702.png" />. The first hypercentre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048370/h0483703.png" /> is the centre of the group (cf. [[Centre of a group|Centre of a group]]); if all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048370/h0483704.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048370/h0483705.png" />, are known, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048370/h0483706.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048370/h0483707.png" /> is a limit ordinal number; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048370/h0483708.png" /> is the complete pre-image of the centre of the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048370/h0483709.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048370/h04837010.png" /> is a non-limit ordinal number. The hypercentres of a group are locally nilpotent.



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J.S. Robinson,   "Finiteness condition and generalized soluble groups" , '''1–2''' , Springer  (1972)</TD></TR></table>