Identity of Group is in Singleton Conjugacy Class
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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