# Second center

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This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
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## Definition

### 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 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 inner automorphisms.
3. It is the second member of the upper central series of $G$.

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