Theorem 4.17

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Statement:

In a group $G$ with binary operation $*$, there is only one element $e$ in $G$ such that

(1)

\begin{equation} e * x = x * e = x \end{equation}

for all $x \in G$. Likewise for each $a \in G$, there is only one element $a'$ in $G$ such that

(2)

\begin{equation} a' * a = a * a' = e \end{equation}

In summary, the identity element and inverse of each element are unique in a group.

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Proof:

Theorem 3.13 shows that an identity element for any binary structure is unique. No use of the group axioms was required to show this.
Turning to the uniqueness of an inverse, suppose that $a \in G$ has inverses $a'$ and $a''$ so that $a' * a = a * a' = e$ and $a'' * a = a * a'' = e$. Then

(3)

\begin{equation} a * a'' = a * a' = e \end{equation}

and by Theorem 4.15,

(4)

\begin{equation} a'' = a', \end{equation}

so the inverse of a group is unique.

Note that in a group $G$, we have

(5)

\begin{equation} (a * b) * (b' * a') = a * (b * b') * a' = (a * e) * a' = a * a' = e. \end{equation}

This equation and Theorem 4.17 show that $b' * a'$ is the unique inverse of $a * b$.

That is, $(a * b)' = b' * a'$. We state this as a corollary(see Corollary 4.18).