Function extension expression

This page describes a formal expression, or formalism, that can be used to describe certain subgroup properties.

View a complete list of formal expressions for subgroup properties OR [[:Category:{{{1}}}|View subgroup properties expressible using this formalism]]

Definition

Main definition

A function restriction expression is the expression of a subgroup property in terms of two properties of functions (by function here is meant a function from a group to itself). The function extension expression corresponding to function properties $a$ and $b$ is denoted as:

$a \leftarrow b$

meaning that $H$ satisfies the property in $G$ if every function $f$ satisfying $b$ on $H$ , there exists a function $f^{'}$ satisfying $a$ in $G$ , and such that $f^{'} |_{H} = f$ .

The property on the left of the arrow is termed the left side of the function restriction expression, and the property on the right side of the arrow is termed the right side of the function restriction expression.