Vipul: Created page with "{{in-group subgroup property|powering-invariant normal subgroup|nilpotent group}} {{in-group subgroup property|quotient-powering-invariant subgroup|nilpotent group}} ==Defini..."

2013-03-01T17:12:38Z

Created page with "{{in-group subgroup property|powering-invariant normal subgroup|nilpotent group}} {{in-group subgroup property|quotient-powering-invariant subgroup|nilpotent group}} ==Defini..."

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{{in-group subgroup property|powering-invariant normal subgroup|nilpotent group}}
{{in-group subgroup property|quotient-powering-invariant subgroup|nilpotent group}}

==Definition==

A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed a '''powering-invariant normal subgroup of nilpotent group''' if it satisfies the following equivalent conditions:

# <math>G</math> is a [[nilpotent group]] and <math>H</math> is a [[powering-invariant normal subgroup]] of <math>G</math>, i.e., <math>H</math> is a [[normal subgroup]] and is a [[powering-invariant subgroup]] of <math>G</math>. Here, by ''powering-invariant'', we mean if <math>p</math> is a [[prime number]] such that <math>G</math> is <math>p</math>-[[powered group for a set of primes|powered]], <math>H</matH> is also <math>p</math>-powered.
# <math>G</math> is a [[nilpotent group]] and <math>H</math> is a [[quotient-powering-invariant subgroup]] of <math>G</math>, i.e., <math>H</math> is a [[normal subgroup]] and if <math>p</math> is a [[prime number]] such that <math>G</math> is <math>p</math>-[[powered group for a set of primes|powered]], the [[quotient group]] <math>G/H</math> is also <math>p</math>-powered.

===Equivalence of definitions===

* (1) implies (2) follows from [[normal subgroup of nilpotent group satisfies the subgroup-to-quotient powering-invariance implication]].
* (2) implies (1) follows from [[quotient-powering-invariant implies powering-invariant]].

Powering-invariant normal subgroup of nilpotent group - Revision history

Vipul: Created page with "{{in-group subgroup property|powering-invariant normal subgroup|nilpotent group}} {{in-group subgroup property|quotient-powering-invariant subgroup|nilpotent group}} ==Defini..."