Left residual operator for composition

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This is a binary subgroup property operator, viz an operator that takes as input two subgroup properties, and outputs one subgroup property

Definition

Property-theoretic statement

Suppose p,q are two subgroup properties. The left residual of p by q is the unique subgroup property r such that:

s * q \le p \iff s \le r.

Here, * denotes the composition operator.

Statement with symbols

Suppose p,q are two subgroup properties. The left residual of p by q is defined as the subgroup property r as follows:

H has property r in G if whenever G is embedded as a subgroup with property q in a group K, H has property p in K.

Facts

Relation with left transiter

Further information: Left transiter

The left transiter of a subgroup property p is defined as the left residual of p by itself.