Inverse of Identity Element is Itself

Theorem

Let $\struct {S, \circ}$ be an algebraic structure with an identity element $e$.

Let the inverse of $e$ be $e^{-1}$.

Then:

$e^{-1} = e$

That is, $e$ is self-inverse.

Proof

From Identity Element is Idempotent:

$e \circ e = e$

Hence the result by definition of identity element.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.6$. Elementary theorems on groups: Example $85$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 31$ Identity element and inverses
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 35.4$: Elementary consequences of the group axioms