# Second center

From Groupprops

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 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.