Identity-true subgroup property
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
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
How they came about
A subgroup property is termed identity-true if it is implied by the improper property or the identity property (for the composition operator), or equivalently, if every group satisfies the property as a subgroup of itself. In other words, a subgroup property is identity-true if ≤ .
Definition with symbols
A subgroup property is termed identity-true if any group satisfies as a subgroup of itself.
Stability under binary operators
The following are true:
- The composition operator applied to identity-true properties is again an identity-true property.
- The intersection operator applied to identity-true properties is again an identity-true property.
- The subgroup generation operator applied to identity-true properties is again an identity-true property.
Thus, for each of these, the identity-true properties forms a submonoid of the corresponding monoid.
Relation with transitivity
Being identity-true is one of the two conditions for being a t.i. subgroup property, and the latter is one of the most important subgroup metaproperties. The other condition is that of being transitive. The relation and interplay of these properties is captured somewhat in the residuation master theorem and the transiter master theorem.