# Group with Zero Element is Trivial

## Theorem

Let $\struct {G, \circ}$ be a group.

Let $\struct {G, \circ}$ have a zero element.

Then $\struct {G, \circ}$ is the trivial group.

## Proof

Let $e \in G$ be the identity element of $G$.

Let $z \in G$ be a zero element.

Let $x \in G$ be any arbitrary element of $\struct {G, \circ}$.

Then:

 $\displaystyle x$ $=$ $\displaystyle x \circ e$ $\quad$ Group Axiom $G \, 2$: Identity $\quad$ $\displaystyle$ $=$ $\displaystyle x \circ \paren {z \circ z^{-1} }$ $\quad$ Group Axiom $G \, 3$: Inverses $\quad$ $\displaystyle$ $=$ $\displaystyle \paren {x \circ z} \circ z^{-1}$ $\quad$ Group Axiom $G \, 1$: Associativity $\quad$ $\displaystyle$ $=$ $\displaystyle z \circ z^{-1}$ $\quad$ Definition of Zero Element: $x \circ z = z$ $\quad$ $\displaystyle$ $=$ $\displaystyle e$ $\quad$ Group Axiom $G \, 3$: Inverses $\quad$

So whatever $x \in G$ is, it has to be the identity element of $G$.

So $G$ can contain only that one element, and is therefore the trivial group.

$\blacksquare$