# Trivial Subgroup is Normal

## Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Then the trivial subgroup $\struct {\set e, \circ}$ of $G$ is a normal subgroup in $G$.

## Proof

First we note that the trivial group $\struct {\set e, \circ}$ is a subgroup of $G$.

To show $\struct {\set e, \circ}$ is normal in $G$:

$\forall a \in G: a \circ e \circ a^{-1} = a \circ a^{-1} = e$

Hence the result.

$\blacksquare$