Finite-p-potentially characteristic subgroup: Difference between revisions

Latest revision as of 22:13, 19 October 2009

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Statement

Suppose $p$ is a prime number and $G$ is a finite $p$ -group (i.e., a group of prime power order). In other words, $G$ is a group of prime power order. A subgroup $H$ of $G$ is termed a finite- $p$ -potentially characteristic subgroup if there exists a finite $p$ -group $K$ containing $G$ such that $H$ is a characteristic subgroup of $K$ .

Facts

Every finite p-group is a subgroup of a finite p-group that is not characteristic in any finite p-group properly containing it

Relation with other properties

The generalization of this property to finite groups, rather than just finite $p$ -groups, is the property of being a finite-pi-potentially characteristic subgroup.

Stronger properties

Characteristic subgroup of group of prime power order
Central subgroup of group of prime power order
Cyclic normal subgroup of group of prime power order
Homocyclic normal subgroup of group of prime power order

Weaker properties

Normal subgroup of group of prime power order

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+{{subgroup property}}
 ==Statement==
-Suppose <math>p</math> is a [[prime number]] and <math>G</math> is a finite <math>p</math>-group. In other words, <math>G</math> is a [[group of prime power order]]. A subgroup <math>H</math> of <math>G</math> is termed a '''<math>p</math>-finite-potentially characteristic subgroup''' if there exists a finite <math>p</math>-group <math>K</math> containing <math>G</math> such that <math>H</math> is a [[characteristic subgroup]] of <math>K</math>.
+Suppose <math>p</math> is a [[prime number]] and <math>G</math> is a finite <math>p</math>-group (i.e., a [[group of prime power order]]). In other words, <math>G</math> is a [[group of prime power order]]. A subgroup <math>H</math> of <math>G</math> is termed a '''finite-<math>p</math>-potentially characteristic subgroup''' if there exists a finite <math>p</math>-group <math>K</math> containing <math>G</math> such that <math>H</math> is a [[characteristic subgroup]] of <math>K</math>.
 ==Facts==
 * [[Every finite p-group is a subgroup of a finite p-group that is not characteristic in any finite p-group properly containing it]]
+==Relation with other properties==
+The generalization of this property to [[finite group]]s, rather than just finite <math>p</math>-groups, is the property of being a [[finite-pi-potentially characteristic subgroup]].
+===Stronger properties===
+* [[Weaker than::Characteristic subgroup of group of prime power order]]
+* [[Central subgroup]] of group of prime power order
+* [[Cyclic normal subgroup]] of group of prime power order
+* [[Homocyclic normal subgroup]] of group of prime power order
+===Weaker properties===
+* [[Stronger than::Normal subgroup of group of prime power order]]