Upper join-closed subgroup property

From Groupprops
Revision as of 11:21, 26 March 2007 by Vipul (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
View a complete list of subgroup metaproperties
View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions
This article is about a general term. A list of important particular cases (instances) is available at Category:Upper join-closed subgroup properties

Definition

Definition with symbols

A subgroup property p is said to be upper join-closed if whenever H \le G and K_1,K_2 are intermediate subgroups of G containing H, then:

H satisfies p in K_1 and H satisfies p in K_2 \implies H satisfies p in <K_1,K_2>.

Relation with other metaproperties

Stronger metaproperties

Related notions

Given a subgroup property p that is identity-true, upper join-closed and also satisfies the intermediate subgroup condition, we can, given any subgroup H of G associate a unique largest subgroup M containing H for which H satisfies p in M.

Such a subgroup property is termed an izable subgroup property and the M that we get is termed the izing subgroup of H for that subgroup property.

Properties satisfying it

Normality

Normality is an upper join-closed subgroup property, viz, if H \le G and K_1, K_2 are intermediate subgroups such that Failed to parse (unknown function "\triangeleleft"): H \triangeleleft K_1 and H \triangleleft K_2, then H \triangleleft <K_1,K_2>.

Central factor

The property of being a central factor is also upper join-closed, in fact, it is izable.