Conjugacy Class of Identity is only Conjugacy Class which is Subgroup

Theorem

Let $G$ be a group.

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

Let $\conjclass g$ denote the conjugacy class of the element $g$.

Then conjugacy class of identity is the only conjugacy class which is a subgroup of $G$:

$\conjclass g < G \iff g = e$

Proof

Necessary Condition

Assume $g = e$.

Then by Identity of Group is in Singleton Conjugacy Class, $\conjclass e = \set e$, which is the trivial subgroup.

$\Box$

Sufficient Condition

Assume $g \neq e$.

Then by Conjugacy Classes are Disjoint, $e \notin \conjclass g$.

Since $\conjclass g$ does not contain the identity element, it could not be a subgroup.

$\blacksquare$

Sources

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