Second derived subgroup

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This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

Definition

Definition with symbols

The second derived subgroup of a group G, denoted \! G'', is defined in the following equivalent ways:

  1. It is the subgroup generated by all elements of the form [[x,y],[z,w]], where [,] denotes the commutator and x,y,z,w \in G.
  2. It is the normal closure of the subgroup generated by all elements of the form [[x,y],[z,w]] where [,] denotes the commutator and x,y,z,w \in G.
  3. It is the derived subgroup of the derived subgroup of G.
  4. It is the intersection of all subgroups H of G for which G/H is a metabelian group, i.e., a solvable group of derived length two.