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Difference between revisions of "Hypercentre"

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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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A member $Z_\alpha$ of the transfinite [[upper central series]] of a group $G$. The first hypercentre $Z_1$ is the [[Centre of a group|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.
  
  

Revision as of 18:19, 1 January 2015

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.


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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=17756
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article