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Central series of a group

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

for which lies in the centre of for all (see also Subgroup series). If for all the subgroup is the complete centre of , then the series is called the upper central series of and if the commutator subgroup of and coincides with , then the series is called the lower central series of .

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.


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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=31982
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article