Subnormal subgroup

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, in the field of <a href="/facts/Group_theory/NfowcI5H">group theory</a>, a <a href="/facts/Subgroup/r91mBgpa">subgroup</a> H of a given <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one <a href="/facts/Normal_subgroup/aRDaGwLm">normal</a> in the next, beginning at H and ending at G.
In notation, 
 
 
 
 H
 
 
 {\displaystyle H}
 
 is 
 
 
 
 k
 
 
 {\displaystyle k}
 
-subnormal in 
 
 
 
 G
 
 
 {\displaystyle G}
 
 if there are subgroups

H
        =
        
          H
          
            0
          
        
        ,
        
          H
          
            1
          
        
        ,
        
          H
          
            2
          
        
        ,
        …
        ,
        
          H
          
            k
          
        
        =
        G
      
    
    {\displaystyle H=H_{0},H_{1},H_{2},\ldots ,H_{k}=G}

of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 such that 
 
 
 
 
 H
 
 i
 
 
 
 
 {\displaystyle H_{i}}
 
 is normal in 
 
 
 
 
 H
 
 i
 +
 1
 
 
 
 
 {\displaystyle H_{i+1}}
 
 for each 
 
 
 
 i
 
 
 {\displaystyle i}
 
.
A subnormal subgroup is a subgroup that is 
 
 
 
 k
 
 
 {\displaystyle k}
 
-subnormal for some positive integer 
 
 
 
 k
 
 
 {\displaystyle k}
 
.
Some facts about subnormal subgroups:

<ul><li>A 1-subnormal subgroup is a proper normal subgroup (and vice versa).</li>
<li>A <a href="/facts/Finitely_generated_group/fVsFHrmb">finitely generated group</a> is <a href="/facts/Nilpotent_group/BJuochky">nilpotent</a> if and only if each of its subgroups is subnormal.</li>
<li>Every <a href="/facts/Quasinormal_subgroup/twYaQCf0">quasinormal subgroup</a>, and, more generally, every <a href="/facts/Conjugate-permutable_subgroup/9P5EuMMU">conjugate-permutable subgroup</a>, of a finite group is subnormal.</li>
<li>Every <a href="/facts/Pronormal_subgroup/bbF3Fbod">pronormal subgroup</a> that is also subnormal, is normal. In particular, a <a href="/facts/Sylow_subgroup/aVTxDRUo">Sylow subgroup</a> is subnormal if and only if it is normal.</li>
<li>Every 2-subnormal subgroup is a conjugate-permutable subgroup.</li></ul>
The property of subnormality is <a href="/facts/Transitive_relation/9r90y50r">transitive</a>, that is, a subnormal subgroup of a subnormal
subgroup is subnormal. The relation of subnormality can be defined as the <a href="/facts/Transitive_closure/HEtgYGM4">transitive closure</a> of the relation of normality.
If every subnormal subgroup of G is normal in G, then G is called a <a href="/facts/T-group_(mathematics)/285Gkuvq">T-group</a>.

Subnormal subgroup open-in-new

Subnormal subgroup