Second center

Definition with symbols
The second center of a group $$G$$, denoted $$Z_2(G)$$, is defined in the following equivalent ways:


 * 1) It is the subgroup $$H$$ of $$G$$ such that $$H$$ contains the defining ingredient::center $$Z(G)$$ of $$G$$, and $$H/Z(G)$$ is the center of the quotient group $$G/Z(G)$$.
 * 2) It is the set of all elements $$h \in G$$ such that conjugation by $$h$$ commutes with conjugation by $$g$$ for every $$g \in G$$. In other words, it is the subgroup comprising the elements whose induced inner automorphisms centralize all defining ingredient::inner automorphisms.
 * 3) It is the second member of the defining ingredient::upper central series of $$G$$.

For more about the properties satisfied and not satisfied by this, see upper central series.