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Difference between revisions of "Central series of a group"

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A [[Normal series|normal series]] all factors of which are central, that is, a series of subgroups
 
A [[Normal series|normal series]] all factors of which are central, that is, a series of subgroups
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021200/c0212001.png" /></td> </tr></table>
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$$E=G_0\subseteq G_1\subseteq\dotsb\subseteq G_n=G$$
  
for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021200/c0212002.png" /> lies in the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021200/c0212003.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021200/c0212004.png" /> (see also [[Subgroup series|Subgroup series]]). If for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021200/c0212005.png" /> the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021200/c0212006.png" /> is the complete centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021200/c0212007.png" />, then the series is called the upper central series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021200/c0212008.png" /> and if the commutator subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021200/c0212009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021200/c02120010.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021200/c02120011.png" />, then the series is called the lower central series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021200/c02120012.png" />.
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for which $G_{i+1}/G_i$ lies in the centre of $G/G_i$ for all $i$ (see also [[Subgroup series|Subgroup series]]). If for all $i$ the subgroup $G_{i+1}/G_i$ is the complete centre of $G/G_i$, then the series is called the upper central series of $G$ and if the commutator subgroup of $G_{i+1}$ and $G$ coincides with $G_i$, then the series is called the lower central series of $G$.
  
 
A group having a central series is called a [[Nilpotent group|nilpotent group]]. In a nilpotent group the lower and the upper central series have the same length, which is the minimal length of a central series of the group. This length is called the nilpotency class (or the degree of nilpotency) of the group.
 
A group having a central series is called a [[Nilpotent group|nilpotent group]]. In a nilpotent group the lower and the upper central series have the same length, which is the minimal length of a central series of the group. This length is called the nilpotency class (or the degree of nilpotency) of the group.

Latest revision as of 12:34, 14 February 2020

A normal series all factors of which are central, that is, a series of subgroups

$$E=G_0\subseteq G_1\subseteq\dotsb\subseteq G_n=G$$

for which $G_{i+1}/G_i$ lies in the centre of $G/G_i$ for all $i$ (see also Subgroup series). If for all $i$ the subgroup $G_{i+1}/G_i$ is the complete centre of $G/G_i$, then the series is called the upper central series of $G$ and if the commutator subgroup of $G_{i+1}$ and $G$ coincides with $G_i$, then the series is called the lower central series of $G$.

A group having a central series is called a nilpotent group. In a nilpotent group the lower and the upper central series have the same length, which is the minimal length of a central series of the group. This length is called the nilpotency class (or the degree of nilpotency) of the group.


Comments

References

[a1] P. Hall, "The theory of groups" , Macmillan (1959) pp. Chapt. 10
How to Cite This Entry:
Central series of a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Central_series_of_a_group&oldid=13788
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article