Transitive subgroup property
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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
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A subgroup property is termed transitive if, whenever such that:
- satisfies as a subgroup of , and
- satisfies as a subgroup of ,
Then satisfies as a subgroup of .
Definition in terms of the composition operator
If is the composition operator on subgroup properties, then a property is transitive if .
Related survey articles
The following survey articles discuss transitivity:
- Proving transitivity
- Disproving transitivity
- Using transitivity to prove subgroup property satisfaction
Relation with other metaproperties
|Metaproperty||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|t.i. subgroup property||This is the conjunction of being transitive with being identity-true: in other words, a subgroup property is t.i. if it is transitive, and is always satisfied by any group as a subgroup of itself.|
|balanced subgroup property||These are special kinds of transitive subgroup properties that can be expressed via function restriction expressions with both sides being the same.|
|idempotent subgroup property||This is a subgroup property such that .|
|Left-hereditary subgroup property||Any subgroup of a subgroup with this property has this property.|
|Metametaproperty name||Satisfied?||Proof||Statement with symbols|
|conjunction-closed subgroup metaproperty||Yes||The conjunction (AND) of a collection of subgroup properties, each of which is transitive, is also transitive. A corollary of this is that given any subgroup property, there is a strongest transitive subgroup property implied by it; that subgroup property is given by the subordination operator.|
|disjunction-closed subgroup metaproperty||Yes||The disjunction (OR) of two transitive subgroup properties need not be transitive. In other words, it is possible to find subgroup properties that are both transitive but such that the property is not transitive.|
Operators to make a subgroup property transitive
There are three general ways to pass from a general subgroup property to a transitive variation (The term variation could be misleading, as we shall see). Each of these is an idempotent operator and the fixed point space is precisely the space of t.i. subgroup properties. These are:
|left transiter|| operator takes a subgroup property and returns the maximum subgroup property such that . In other words, satisfies property in if whenever has property in , also has property in .|
It turns out, from the transiter master theorem, that the left transiter of any subgroup property is a t.i. subgroup property (that is, both transitive and identity-true), and further, that the left transiter of a t.i. subgroup property is itself.
|right transiter||operator takes a subgroup property and returns the maximum subgroup property such that . In other words, satisfies property in if and only if for every subgroup satisfying in , must also satisfy in .|
|subordination operator (we can think of this as the Kleene star closure with respect to the composition operator)||The subordination of a property is the property of being a subgroup such that there is a finite length chain from the subgroup to the whole group wherein each has property in its successor. Note that we allow a chain of length zero.|
Other ways are:
Effect of property modifiers on transitivity
Transfer condition operator
Further information: Transfer condition operator preserves transitivity
Let be a subgroup property. The transfer-closure of is defined as the following subgroup property : A subgroup has property in if has property in for any subgroup of .
Then, if is transitive, so is the transfer-closure of .
The intermediately operator may not in general preserve transitivity.
One can write code to test this subgroup metaproperty in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
View the GAP code for testing this subgroup metaproperty with given biggest group at: IsTransitiveWithBigGroupView the GAP code for testing this subgroup metaproperty on all groups living inside a given big group at: IsTransitiveInAllSubgroupsOfGroup
View other GAP-codable subgroup metaproperties
It is possible to check, given a group and a subgroup property , whether whenever are subgroups such that satisfies property in and satisfies property in , also satisfies property in . Although there is no in-built command for this, it can be achieved using a short snippet of code, available at GAP:IsTransitiveWithBigGroup. This is then used as follows:
We can also check whether, for a given property and a group , whenever are such that satisfies property in and satisfies property in , then satisfies property in . The short snippet of code needed for this is available at GAP:IsTransitiveInAllSubgroupsOfGroup. It is used as follows: