Characteristic-potentially characteristic subgroup: Difference between revisions

Revision as of 18:23, 7 February 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]
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This is a variation of characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity

Definition

Symbol-free definition

A subgroup of a group is termed strongly potentially characteristic if there is an embedding of the bigger group in some group such that, in that embedding both the group and the subgroup become characteristic.

Definition with symbols

A subgroup $H$ of a group $G$ is termed potentially characteristic in $G$ if there exists a group $K$ containing $G$ such that both $H$ and $G$ are characteristic in $K$ .

In terms of the strongly potentially operator

The subgroup property of being potentially characteristic is obtained by applying the strongly potentially operator to the subgroup property of being characteristic.

Relation with other properties

Stronger properties

Characteristic subgroup

Weaker properties

Conjecture of equalling normality

This property is conjectured to equal the property: normality

The NSPC conjecture states that every normal subgroup is strongly potentially characteristic. In other words, if $H ◃ G$ , there is a group $K$ containing $G$ such that both both $H$ and $G$ are characteristic in $K$ .

Metaproperties

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
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Intersection-closedness

The problem of whether an intersection (finite or arbitrary) of subgroups with this property again has this property is an open problem.

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Property operators

Left transiter

Every characteristic subgroup of a strongly potentially characteristic subgroup is strongly potentially characteristic. In fact, the same supergroup works.

That is, suppose $M \leq G \leq H$ with $M$ characteristic in $G$ and $G$ strongly potentially characteristic in $H$ . Then, there exists a group $K$ containing $H$ such that both $G$ and $H$ are characteristic in $K$ . Then, we also have that both $M$ and $H$ are characteristic in $K$ , and hence $M$ is strongly potentially characteristic in $H$ .

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 {{conjecturedtoequal|normality}}
-The [[NSPC conjecture]] states that every normal subgorup is strongly potentially characteristic. In other words, if <math>H \triangleleft G</math>, there is a group <math>K</math> containing <math>G</math> such that both both <math>H</math> and <math>G</math> are characteristic in <math>K</math>.
+The [[NSPC conjecture]] states that every normal subgroup is strongly potentially characteristic. In other words, if <math>H \triangleleft G</math>, there is a group <math>K</math> containing <math>G</math> such that both both <math>H</math> and <math>G</math> are characteristic in <math>K</math>.
 ==Metaproperties==