Epimorphism Preserves Groups

Theorem

Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures.

Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an epimorphism.

If $\left({S, \circ}\right)$ is a group, then so is $\left({T, *}\right)$.

Proof

• From Epimorphism Preserves Semigroups, if $\left({S, \circ}\right)$ is a semigroup then so is $\left({T, *}\right)$.

• From Epimorphism Preserves Identity, if $\left({S, \circ}\right)$ has an identity $e_S$, then $\phi \left({e_S}\right)$ is the identity for $*$.

• From Epimorphism Preserves Inverses, if $x^{-1}$ is an inverse of $x$ for $\circ$, then $\phi \left({x^{-1}}\right)$ is an inverse of $\phi \left({x}\right)$ for $*$.

The result follows from the definition of group.

$\blacksquare$

Also see

• Isomorphism Preserves Groups

• Epimorphism Preserves Associativity
• Epimorphism Preserves Commutativity
• Epimorphism Preserves Identity
• Epimorphism Preserves Inverses

• Epimorphism Preserves Semigroups

• Epimorphism Preserves Distributivity

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 12$: Theorem $12.2$: Corollary