Group extension problem: Difference between revisions

Revision as of 17:01, 10 July 2008

Statement

The group extension problem for two groups $N$ and $H$ , is the problem of finding all groups $G$ with $N$ as a normal subgroup of $G$ , and the quotient group $G / N$ isomorphic to $H$ .

Congruence classes formulation

In this formulation, we're thinking of $N$ and $H$ as specific groups, and looking at short exact sequences:

$1 \to N \to G \to H \to 1$

Formulation upto automorphisms

Classifying group extensions for an Abelian normal subgroup

If $N$ is an Abelian group, then there is a procedure to classify all group extensions with normal subgroup $N$ and

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 The '''group extension problem''' for two groups <math>N</math> and <math>H</math>, is the problem of finding all groups <math>G</math> with <math>N</math> as a normal subgroup of <math>G</math>, and the quotient group <math>G/N</math> isomorphic to <math>H</math>.
-More specifically, we are interested in finding all '''congruence classes''' of extensions, where <math>G_1</math> is congruent to <math>G_2</math> if there is an isomorphism of the groups that sends <math>N</math> to itself via the identity map.
+===Congruence classes formulation===
+In this formulation, we're thinking of <math>N</math> and <math>H</math> as specific groups, and looking at ''short exact sequences'':
+<math>1 \to N \to G \to H \to 1</math>
+===Formulation upto automorphisms===
 ==Classifying group extensions for an Abelian normal subgroup==
 If <math>N</math> is an [[Abelian group]], then there is a procedure to classify all group extensions with normal subgroup <math>N</math> and