Transitive subgroup property

Definition
A subgroup property $$p$$ is termed transitive if, whenever $$H \le K \le G$$ such that:


 * $$H$$ satisfies $$p$$ as a subgroup of $$K$$, and
 * $$K$$ satisfies $$p$$ as a subgroup of $$G$$,

Then $$H$$ satisfies $$p$$ as a subgroup of $$G$$.

Definition in terms of the composition operator
If $$*$$ is the composition operator on subgroup properties, then a property $$p$$ is transitive if $$p * p \le p$$.

Related survey articles
The following survey articles discuss transitivity:


 * Proving transitivity
 * Disproving transitivity
 * Using transitivity to prove subgroup property satisfaction

Opposite metaproperties

 * Antitransitive subgroup property

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:

Other ways are:


 * ascendant closure
 * descendant closure
 * serial closure

Transfer condition operator
Let $$p$$ be a subgroup property. The transfer-closure of $$p$$ is defined as the following subgroup property $$q$$: A subgroup $$H$$ has property $$q$$ in $$G$$ if $$H \cap K$$ has property $$p$$ in $$K$$ for any subgroup $$K$$ of $$G$$.

Then, if $$p$$ is transitive, so is the transfer-closure of $$p$$.

Intermediately operator
The intermediately operator may not in general preserve transitivity.

Testing
It is possible to check, given a group $$G$$ and a subgroup property $$p$$, whether whenever $$H \le K \le G$$ are subgroups such that $$H$$ satisfies property $$p$$ in $$K$$ and $$K$$ satisfies property $$p$$ in $$G$$, $$H$$ also satisfies property $$p$$ in $$G$$. 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:

IsTransitiveWithBigGroup(Group,Property)

We can also check whether, for a given property $$p$$ and a group $$G$$, whenever $$H \le K \le L \le G$$ are such that $$H$$ satisfies property $$p$$ in $$K$$ and $$K$$ satisfies property $$p$$ in $$L$$, then $$H$$ satisfies property $$p$$ in $$L$$. The short snippet of code needed for this is available at GAP:IsTransitiveInAllSubgroupsOfGroup. It is used as follows:

IsTransitiveInAllSubgroupsOfGroup(Group,Property)