Set theory of the real line: Difference between revisions

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In [[mathematics]], in the field of [[group theory]], a '''metanilpotent group''' is a group that is nilpotent by nilpotent. In other words, it has a normal [[nilpotent group|nilpotent]] subgroup such that the quotient group is also nilpotent.
 
In symbols, <math>G</math> is metanilpotent if there is a [[normal subgroup]] <math>N</math> such that both <math>N</math> and <math>G/N</math> are nilpotent.
 
The following are clear:
 
* Every metanilpotent group is a [[solvable group]].
* Every subgroup and every quotient of a metanilpotent group is metanilpotent.
 
==References==
* J.C. Lennox, D.J.S. Robinson, ''The Theory of Infinite Soluble Groups'', [[Oxford University Press]], 2004, ISBN 0-19-850728-3. P.27.
* D.J.S. Robinson, ''A Course in the Theory of Groups'', GTM '''80''', [[Springer Verlag]], 1996, ISBN 0-387-94461-3. P.150.
 
[[Category:Group theory]]
[[Category:Solvable groups]]
[[Category:Properties of groups]]

Latest revision as of 14:24, 16 December 2013

In mathematics, in the field of group theory, a metanilpotent group is a group that is nilpotent by nilpotent. In other words, it has a normal nilpotent subgroup such that the quotient group is also nilpotent.

In symbols, is metanilpotent if there is a normal subgroup such that both and are nilpotent.

The following are clear:

  • Every metanilpotent group is a solvable group.
  • Every subgroup and every quotient of a metanilpotent group is metanilpotent.

References

  • J.C. Lennox, D.J.S. Robinson, The Theory of Infinite Soluble Groups, Oxford University Press, 2004, ISBN 0-19-850728-3. P.27.
  • D.J.S. Robinson, A Course in the Theory of Groups, GTM 80, Springer Verlag, 1996, ISBN 0-387-94461-3. P.150.
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