Difference between revisions of "Subgroup with abelianization of maximum order"

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{{subgroup property}}
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{{abelian maximality notion in p-groups}}
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==Definition==
 
==Definition==
  
 
Let <math>P</math> be a [[group of prime power order]]. A [[subgroup]] <math>B</math> of <math>P</math> is termed a '''subgroup with abelianization of maximum order''' if the [[order of a group|order]] of the [[abelianization]] of <math>B</math> (i.e., the quotient of <math>B</math> by its [[commutator subgroup]]) is greater than or equal to the order of the abelianization of any subgroup of <math>P</math>.
 
Let <math>P</math> be a [[group of prime power order]]. A [[subgroup]] <math>B</math> of <math>P</math> is termed a '''subgroup with abelianization of maximum order''' if the [[order of a group|order]] of the [[abelianization]] of <math>B</math> (i.e., the quotient of <math>B</math> by its [[commutator subgroup]]) is greater than or equal to the order of the abelianization of any subgroup of <math>P</math>.
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==Relation with other properties==
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===Stronger properties===
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* [[Weaker than::Minimal subgroup with abelianization of maximum order]]

Latest revision as of 22:18, 29 July 2009

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
This article is about a maximality notion among subgroups, related to abelianness or small class, in a group of prime power order.
View other such notions

Definition

Let P be a group of prime power order. A subgroup B of P is termed a subgroup with abelianization of maximum order if the order of the abelianization of B (i.e., the quotient of B by its commutator subgroup) is greater than or equal to the order of the abelianization of any subgroup of P.

Relation with other properties

Stronger properties