Subgroup series

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, specifically <a href="/facts/Group_theory/NfowcI5H">group theory</a>, a subgroup series of a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is a <a href="/facts/Chain_(order_theory)/xnTtmuYJ">chain</a> of <a href="/facts/Subgroup/r91mBgpa">subgroups</a>:

1
        =
        
          A
          
            0
          
        
        ≤
        
          A
          
            1
          
        
        ≤
        ⋯
        ≤
        
          A
          
            n
          
        
        =
        G
      
    
    {\displaystyle 1=A_{0}\leq A_{1}\leq \cdots \leq A_{n}=G}

where 
 
 
 
 1
 
 
 {\displaystyle 1}
 
 is the <a href="/facts/Trivial_group/XPjQkQFm">trivial subgroup</a>. Subgroup series can simplify the study of a group to the study of simpler subgroups and their relations, and several subgroup series can be invariantly defined and are important invariants of groups. A subgroup series is used in the <a href="/facts/Subgroup_method/GyPLlS0e">subgroup method</a>.
Subgroup series are a special example of the use of <a href="/facts/Filtration_(mathematics)/qcvLvKRI">filtrations</a> in <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>.

Subgroup series open-in-new

Subgroup series