Normal complement: Difference between revisions

Latest revision as of 21:39, 28 February 2009

Definition

Definition with symbols

Suppose $H$ is a subgroup of a group $G$ . A normal complement $N$ to $H$ in $G$ is a normal subgroup of $G$ that is a permutable complement to $H$ in $G$ . In other words, $N \cap H$ is trivial and $N H = G$ .

Every subgroup need not have a normal complement. A subgroup that possesses a normal complement is termed a retract.

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 ===Definition with symbols===
-Suppose <math>H</math> is a [[subgroup]] of a [[group]] <math>G</math>. A '''normal complement''' <math>N</math> to <math>H</math> in <math>G</math> is a [[normal subgroup]] of <math>G</math> that is a [[permutable complement]] to <math>H</math> in <math>G</math>. In other words, <math>N \cap H</math> is trivial and <math>NH = G</math>.
+Suppose <math>H</math> is a [[subgroup]] of a [[group]] <math>G</math>. A '''normal complement''' <math>N</math> to <math>H</math> in <math>G</math> is a [[defining ingredient::normal subgroup]] of <math>G</math> that is a [[defining ingredient::permutable complement]] to <math>H</math> in <math>G</math>. In other words, <math>N \cap H</math> is trivial and <math>NH = G</math>.
+Every subgroup need not have a normal complement. A subgroup that possesses a normal complement is termed a [[defining ingredient::retract]].