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 | ||
| − | + | $$E=G_0\subseteq G_1\subseteq\dotsb\subseteq G_n=G$$ | |
| − | for which | + | 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
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