Composite of Automorphisms is Automorphism

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Theorem

Let $\struct {S, \circ_1, \circ_2, \ldots, \circ_n}$ be an algebraic structure.

Let:

$\phi: \struct {S, \circ_1, \circ_2, \ldots, \circ_n} \to \struct {S, \circ_1, \circ_2, \ldots, \circ_n}$
$\psi: \struct {S, \circ_1, \circ_2, \ldots, \circ_n} \to \struct {S, \circ_1, \circ_2, \ldots, \circ_n}$

be automorphisms.

Then the composite of $\phi$ and $\psi$ is also an automorphism.


Proof

From Composite of Homomorphisms on Algebraic Structure is Homomorphism, $\psi \circ \phi$ is a homomorphism.

By the definition of a composite mapping, $\psi \circ \phi$ is a mapping from $S$ into $S$.

Hence $\psi \circ \phi$ is an automorphism.

$\blacksquare$


Sources