Free quotient group admits a section: Difference between revisions

Latest revision as of 21:38, 16 February 2013

Suppose $N$ is a normal subgroup of a group $G$ such that the quotient group $G / N$ is a free group.

Then, $N$ is a complemented normal subgroup of $G$ . In other words, there exists a retract $B$ of $G$ with normal complement $N$ , i.e., $B$ is a subgroup of $G$ such that $G$ is the internal semidirect product $N ⋊ B$ . Explicitly, $N \cap B$ is trivial and $N B = G$ .

A normal subgroup $N$ such that $G / N$ is a free group is termed a free-quotient subgroup.

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 ==Statement==
-Suppose <math>N</math> is a [[normal subgroup]] of a [[group]] <math>G</math> such that the [[quotient group]] <math>G/N</math> is a [[fact about::free group]].
+Suppose <math>N</math> is a [[normal subgroup]] of a [[group]] <math>G</math> such that the [[quotient group]] <math>G/N</math> is a [[fact about::free group;3| ]][[free group]].
 Then, <math>N</math> is a [[fact about::complemented normal subgroup;2| ]][[complemented normal subgroup]] of <math>G</math>. In other words, there exists a [[retract]] <math>B</math> of <math>G</math> with [[normal complement]] <math>N</math>, i.e., <math>B</math> is a subgroup of <math>G</math> such that <math>G</math> is the [[fact about::internal semidirect product;2| ]][[internal semidirect product]] <math>N \rtimes B</math>. Explicitly, <math>N \cap B</math> is trivial and <math>NB = G</math>.