Right transiter

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This article defines a subgroup property modifier (a unary subgroup property operator) -- viz an operator that takes as input a subgroup property and outputs a subgroup property


View a complete list of subgroup property modifiers OR View a list of all subgroup property operators (possibly with multiple inputs)

This property modifier is idempotent and a property is a fixed-point, or equivalently, an image of this if and only if it is a:t.i. subgroup property

Definition

Symbol-free definition

The right transiter is an operator from the subgroup property space to itself, defined as follows. The right transiter of a subgroup property p is the maximum among all subgroup properties q for which the following holds:

p * q \le p

where * denotes the composition operator on subgroup properties.

Definition with symbols

The right transiter of a subgroup property p is the property q defined by the following criterion:

A subgroup H in a group G is said to have q if and only if whenever K has property p as a subgroup of H, K also has property p in G.

We denote the right transiter of a property p as R(p).

In terms of the residual operator

The right transiter of a property is its right residual by itself, with respect to the composition operator.

A general notion of right transiter can be given for any associative quantalic binary operator on a property space. The generic notion inherits many of the nice behaviours from this special case.

Effect on subgroup metaproperties

Template:Intsubcondn-preserving

If p satisfies the intermediate subgroup condition, so does the right transiter of p.

Computing the right transiter

In the function restriction formalism

If a subgroup property can be expressed by a function restriction formal expression a \to b, thne we can do the following: