---- > [!definition] Definition. ([[group isomorphism]]) > An **isomorphism of [[group]]s** $G_{1}, G_{2}$ is a [[bijection|bijective]] [[group homomorphism|homomorphism]]. > [!justification] > Note that we do not explicitly require the inverse to be a [[group homomorphism|homomorphism]] as part of this definition. That's because it is one automatically: Let $\phi:G_{1} \to G_{2}$ be a [[group homomorphism]] and $\psi:G_{2} \to G_{1}$ satisfy $\psi=\phi ^{-1}$. Compute $\phi\big( \psi(g) \psi(h) \big)=\phi\big( \psi(g) \big) \phi\big( \psi(h) \big)=gh.$ So $\psi(g)\psi(h)=\phi ^{-1}(gh)$. But $\phi ^{-1}=\psi$. \ Compare to in the [[category]] $\mathsf{Top}$, where the appropriate notion of '[[isomorphism]]' ([[homeomorphism]]) mandates the existence of a *structure-preserving* (i.e. [[continuous]]) [[inverse map]]. ^556381 > [!basicexample] > - $(\mathbb{R}, +)$ and $\mathbb{R}_{>0}$ are [[isomorphism|isomorphic]], as witnessed by the exponential map with inverse $\text{log}$. > - $D_{3}$ and $S_{3}$ are [[group isomorphism|isomorphic]]; find a discussion [[symmetric group|here]]. > - Here is a more surprising example: $(\mathbb{R}, +)$ and $(\mathbb{C}, +)$ are [[group isomorphism|isomorphic]], because they have the same dimension as [[vector spaces|vector spaces]] over $\mathbb{Q}$ (but there is nuance here...) ^basic-example > [!basicnonexample] > No two of the [[group|groups]] $(\mathbb{Z}, +)$ ([[cyclic group|the infinite cyclic group]]), $(\mathbb{Q}, +)$, $(\mathbb{R},+)$ are [[group isomorphism|isomorphic]]. Indeed, $(\mathbb{R},+)$ is clearly not [[group isomorphism|isomorphic]] to either of the other two because it is [[uncountably infinite|uncountable]]. And $(\mathbb{Z}, +) \neq (\mathbb{Q}, +)$ because if $\mathbb{Q}$ were [[generating set of a group|generated by]] a single element $q_{0}$, then every rational number would equal $mq_{0}$ for some $m \in \mathbb{Z}$. That cannot happen, e.g. $m\frac{1}{2}=\frac{m}{2} \neq \frac{1}{4}$ for any $m \in \mathbb{Z}$. So $\mathbb{Z}$ is [[cyclic group|cyclic]] while $\mathbb{Q}$ is not, see [[group homomorphisms preserve structure]]. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```