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 with symbols
The second center of a group , denoted , is defined in the following equivalent ways:
- It is the subgroup of such that contains the center of , and is the center of the quotient group .
- It is the set of all elements such that conjugation by commutes with conjugation by for every . In other words, it is the subgroup comprising the elements whose induced inner automorphisms centralize all inner automorphisms.
- It is the second member of the upper central series of .
For more about the properties satisfied and not satisfied by this, see upper central series.