Center (group theory)

The center of a group $G$ (denoted $Z(G)$) is the set of all elements which commute with every element of the group. That is, the center of $G$ is $\{a\in G:ab=ba\ \forall b\in G\}$.

Facts

• If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is a group, the center Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z(G)} is a normal subgroup of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} .
• We have $Z(G)=G$ if and only if $G$ is an abelian group.

External links

• Center on MathWorld