Isomorphism in Abstract Algebra 📂Abstract Algebra

Isomorphism in Abstract Algebra

Definition ¹

For two binary operation structures $\left< S , * \right>$ and $\left< S' , *' \right>$, if there exists a bijective function $\phi : S \to S'$ such that for all $x , y \in S$ $$ \phi (x \ast\ y) = \phi ( x ) *' \phi ( y ) $$ is satisfied, then $\phi$ is called an isomorphism, $S$ and $S'$ are said to be isomorphic, and this is written as $S \simeq S'$.

Explanation

To summarize the definition, if there exists a bijection that preserves the operation, we regard the two as essentially the same. Even beyond the isomorphism discussed in abstract algebra, such maps are extremely important throughout mathematics.

If $\phi$ preserves the operation but is not a bijection, it is called a homomorphism. Likewise, there are many important maps that are not isomorphisms, and there is a great deal of research on them.

The following is a theorem meaning that the identity of the identity element is also preserved under an isomorphism.

Theorem

If $S \simeq S'$ under an isomorphism $\phi$ and $e$ is the identity element of $S$, then $\phi (e)$ is the identity element of $S'$.

Proof

Since $e$ is the identity element of $S$, for $s \in S$ $$ e \ast\ s = s \ast\ e = s $$

$$ \phi ( e \ast\ s ) = \phi ( s \ast\ e ) = \phi ( s ) $$ Since $\phi$ is an isomorphism, for $s' : = \phi (s) \in S'$ $$ \phi ( e ) *' \phi ( s ) = \phi ( s ) *' \phi ( e ) = \phi ( s ) $$ Therefore, $\phi (e)$ is the identity element of $S'$.

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