Left-extensibility-stable subgroup property

From Groupprops
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 subgroup metaproperty is related to, or can be defined, using the following formalism: function restriction formalism


BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article is about a general term. A list of important particular cases (instances) is available at Category:Left-extensibility-stable subgroup properties

Definition

Symbol-free definition

A subgroup property is termed left-extensibility-stable if it can be expressed in the function restriction formalism with the left side being extensibility-stable.

Definition with symbols

A subgroup property p is termed left-extensibility-stable if we can write:

p = a \to b

where a is an extensibility-stable function property.

The above symbols mean that:

  • A subgroup H has property p in a group G if and only if every function from G to itself satisfying property a restricts to a function from H to itself satisfying property b.
  • Property a being extensibility-stable means the following: whenever G \le K are groups,

and f is a function on G satisfying a, then there is a function f' on K satisfying a such that the restriction of f' to G is f.

In terms of the left expressibility operator

The metaproperty of being a left-extensibility-stable subgroup property is obtained by applying the left expressibility operator to the function metaproperty of being extensibilility-stable.

Relation with other metaproperties

Stronger metaproperties

Weaker metaproperties