Difference between revisions of "Maxsensitive subgroup"
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Latest revision as of 23:49, 7 May 2008
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]
Contents
Definition
Symbolfree definition
A subgroup of a group is termed maxsensitive if its intersection with any maximal subgroup of the whole group is either equal to it or a maximal subgroup of it.
Definition with symbols
A subgroup of a group is termed maxsensitive in if, for any maximal subgroup of , the group ∩ is either the whole of or a maximal subgroup of .
In terms of the subgroup intersection restriction formalism
The property of being maxsensitive is the balanced subgroup property in the subgroup intersection restriction formalism corresponding to the subgroup property of being a maximal subgroup.
Metaproperties
Transitivity
This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive  View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup propertiesView a complete list of facts related to transitivity of subgroup properties Read a survey article on proving transitivity
The subgroup property of being maxsensitive is transitive, on account of being a balanced subgroup property with respect to a restriction formalism.
Trimness
This subgroup property is trim  it is both trivially true (true for the trivial subgroup) and identitytrue (true for a group as a subgroup of itself).
View other trim subgroup properties  View other trivially true subgroup properties  View other identitytrue subgroup properties
The subgroup property of being maxsensitive is identitytrue, that is, every group is maxsensitive as a subgroup of itself.
It is also trivially true, that is, the trivial subgroup is always a maxsensitive subgroup.