Finite characteristic series operator: Difference between revisions

Latest revision as of 23:29, 7 May 2008

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

Definition

Definition with symbols

The finite characteristic series operator is a group property modifier that takes as input a group property $p$ and outputs a group property $q$ such that a group $G$ satisfies $q$ if and only if it has a characteristic series $e = H_{0} \leq H_{1} \leq \dots \leq H_{r} = G$ such that each quotient $H_{i} / H_{i - 1}$ satisfies property $p$ .

Relation with other modifiers

Weaker modifiers

Properties

Monotonicity

This group property modifier is monotone, viz if $p \leq q$ are group properties and $f$ is the operator, then $f (p) \leq f (q)$

Ascendance

This group property modifier is ascendant, viz the image of any group property under this modifier is always weaker than the group property we started with

Idempotence

This group property modifier is idempotent, viz applying it twice to a group property has the same effect as applying it once

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 ===Definition with symbols===
-The '''finite characteristic series operator''' is a [[group property modifier]] that takes as input a group property <math>p</math> and outputs a group property <math>q</math> such that a group <math>G</math> satisfies <math>q</math> if and only if it has a [[characteristic series]] <math>e = H_0 \le H_1 \le \ldots le H_r = G</math> such that each quotient <math>H_i/H_{i-1}</math> satisfies property <math>p</math>.
+The '''finite characteristic series operator''' is a [[group property modifier]] that takes as input a group property <math>p</math> and outputs a group property <math>q</math> such that a group <math>G</math> satisfies <math>q</math> if and only if it has a [[characteristic series]] <math>e = H_0 \le H_1 \le \ldots \le H_r = G</math> such that each quotient <math>H_i/H_{i-1}</math> satisfies property <math>p</math>.
 ==Relation with other modifiers==