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# Def/Center (group)

Definition of Center of a group: Suppose that $G$ is a group. The center of $G$ is the subset $Z(G)$, defined by: $$Z(G) = \{ g \in G, \mbox{ such that } \forall h \in G, gh = hg \}.$$ In other words, $g$ is in the center of $G$, if $g$ commutes with every element of $G$.

Since $gh = hg$ if and only if $h g h^{-1} = g$, it follows that:

• An element $g \in G$ is in the center of $G$, if and only if the conjugacy class of $g$ contains only $g$, i.e., has one element.

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Group, Def/Commute

The following statements and definitions logically rely on the material of this page: State/Centers of p-groups are nontrivial, and State/Groups of prime squared order are abelian

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Basic group theory