Identity of Group is in Singleton Conjugacy Class

Theorem

Let $G$ be a group.

Let $e$ be the identity of $G$.

Then $e$ is in its own singleton conjugacy class:

$\conjclass e = \set e$

Proof

From Identity of Group is in Center:

$e \in \map Z G$

where $\map Z G$ is the center of $G$.

From Conjugacy Class of Element of Center is Singleton:

$\conjclass e = \set e$

$\blacksquare$

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conjugacy class