Group extension

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a group extension is a general means of describing a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> in terms of a particular <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a> and <a href="/facts/Quotient_group/HlwpHtjb">quotient group</a>. If 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 and 
 
 
 
 N
 
 
 {\displaystyle N}
 
 are two groups, then 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is an extension of 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 by 
 
 
 
 N
 
 
 {\displaystyle N}
 
 if there is a <a href="/facts/Short_exact_sequence/leUJnwJb">short exact sequence</a>

1
        →
        N
        
        
          
            →
            ι
          
        
        
        G
        
        
          
            →
            π
          
        
        
        Q
        →
        1.
      
    
    {\displaystyle 1\to N\;{\overset {\iota }{\to }}\;G\;{\overset {\pi }{\to }}\;Q\to 1.}

If 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is an extension of 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 by 
 
 
 
 N
 
 
 {\displaystyle N}
 
, then 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is a group, 
 
 
 
 ι
 (
 N
 )
 
 
 {\displaystyle \iota (N)}
 
 is a <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a> of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 and the <a href="/facts/Quotient_group/HlwpHtjb">quotient group</a> 
 
 
 
 G
 
 /
 
 ι
 (
 N
 )
 
 
 {\displaystyle G/\iota (N)}
 
 is <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> to the group 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
. Group extensions arise in the context of the extension problem, where the groups 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 and 
 
 
 
 N
 
 
 {\displaystyle N}
 
 are known and the properties of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 are to be determined. Note that the phrasing "
 
 
 
 G
 
 
 {\displaystyle G}
 
 is an extension of 
 
 
 
 N
 
 
 {\displaystyle N}
 
 by 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
" is also used by some.
Since any <a href="/facts/Finite_group/vv2tvogB">finite group</a> 
 
 
 
 G
 
 
 {\displaystyle G}
 
 possesses a <a href="/facts/Maximal_subgroup/XebALFlp">maximal</a> <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a> 
 
 
 
 N
 
 
 {\displaystyle N}
 
 with <a href="/facts/Simple_group/e5hGF7nT">simple</a> <a href="/facts/Factor_group/HlwpHtjb">factor group</a> 
 
 
 
 G
 
 /
 
 ι
 (
 N
 )
 
 
 {\displaystyle G/\iota (N)}
 
, all finite groups may be constructed as a series of extensions with finite <a href="/facts/Simple_group/e5hGF7nT">simple groups</a>. This fact was a motivation for completing the <a href="/facts/Classification_of_finite_simple_groups/nMwLJf1K">classification of finite simple groups</a>.
An extension is called a central extension if the subgroup 
 
 
 
 N
 
 
 {\displaystyle N}
 
 lies in the <a href="/facts/Center_of_a_group/gqAg6B8I">center</a> of 
 
 
 
 G
 
 
 {\displaystyle G}
 
.

Group extension open-in-new

Group extension