Transitive subgroup description rule

This article defines a property that can be evaluated for a subgroup description rule, viz a rule for describing a subgroup of a group given an encoding

Definition

A subgroup description rule is said to be transitive if it satisfies the following two conditions:

• Suppose $G$ is a group equipped with an encoding and $K$ is a subgroup of $G$ described by means of the subgroup description rule. Then, we can use this to obtain an encoding of $K$
• If $H$ is a subgroup of $K$ described using the subgroup description rule inside $K$, then we can retrieve from this a description of $H$ as a subgroup of $G$, using the subgroup description rule

Further, we require that the algorithms for both be polynomial in the input sizes.