Difference between revisions of "Max-sensitive 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]
A subgroup of a group is termed max-sensitive 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 max-sensitive 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
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 properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity
This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties
The subgroup property of being max-sensitive is identity-true, that is, every group is max-sensitive as a subgroup of itself.
It is also trivially true, that is, the trivial subgroup is always a max-sensitive subgroup.