Difference between revisions of "Hypercentre"
From Encyclopedia of Mathematics
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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. | + | 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 group|locally nilpotent]]. |
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====References==== | ====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> | + | <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> | ||
Latest revision as of 19:06, 12 December 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.
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
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