Normal closure-finding problem: Difference between revisions

Revision as of 20:34, 25 June 2013

This article describes the subgroup operator computation problem for the subgroup operator: normal closure

Description

We are given a group $G$ and a subgroup $H$ specified by some means (typically using a generating set, though we may also specify it using a membership test). The goal is to obtain a description of the normal closure $K = H^{G}$ of $H$ in $G$ by some means (which may be using generating sets or membership tests or both).

Relation with other problems

Problems it reduces to

Normality testing problem: See black-box reduction of normal closure-finding in terms of generating sets to normality testing.

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 ==Description==
-We are given a [[group]] <math>G</math> and a [[subgroup]] <math>H</math> specified by some means (typically using [[generating set]]s, though we may also specify them using membership tests). The goal is to obtain a description of the [[normal closure]] <math>K = H^G</math> of <math>H</math> in <math>G</matH> using the same language (so if <math>G</math> and <math>H</math> are described using generating sets, we want a generating set for <math>K</math>, whereas if <math>G</math> and <math>H</math> are described using membership tests, then <math>K</math> is also to be described using a membership test.
+We are given a [[group]] <math>G</math> and a [[subgroup]] <math>H</math> specified by some means (typically using a [[generating set]], though we may also specify it using a membership test). The goal is to obtain a description of the [[normal closure]] <math>K = H^G</math> of <math>H</math> in <math>G</matH> by some means (which may be using generating sets or membership tests or both).
 ==Relation with other problems==