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Nilpotent Group


A group G is nilpotent if the upper central sequence

 1=Z_0<=Z_1<=Z_2<=...<=Z_n<=...

of the group terminates with Z_n=G for some n.

Nilpotent groups have the property that each proper subgroup is properly contained in its normalizer. A finite nilpotent group is the direct product of its Sylow p-subgroups.


See also

Group Center, Group Upper Central Series, Nilpotent Lie Group

This entry contributed by John Renze

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References

Curtis, C. and Reiner, I. Methods of Representation Theory. New York: Wiley, 1981.

Referenced on Wolfram|Alpha

Nilpotent Group

Cite this as:

Renze, John. "Nilpotent Group." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/NilpotentGroup.html

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