Isomorphism Preserves Groups

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Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an isomorphism.

If $\struct {S, \circ}$ is a group, then so is $\struct {T, *}$.

Proof 1

From Isomorphism Preserves Semigroups, if $\struct {S, \circ}$ is a semigroup then so is $\struct {T, *}$.

From Isomorphism Preserves Identity, if $\struct {S, \circ}$ has an identity $e_S$, then $\map \phi {e_S}$ is the identity for $*$.

From Isomorphism Preserves Inverses, if $x^{-1}$ is an inverse of $x$ for $\circ$, then $\map \phi {x^{-1} }$ is an inverse of $\map \phi x$ for $*$.

The result follows from the definition of group.


Proof 2

An isomorphism is an epimorphism.

The result follows as a direct corollary of Epimorphism Preserves Groups.