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injection can be extended to isomorphism
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(Theorem)
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Theorem. If $f$ is an injection from a set $S$ into a group $G$ , then there exist a group $H$ containing $S$ and a group isomorphism $\varphi\!:H \to G$ such that $\varphi|S = f$ .
Proof. Let $M$ be a set such that $\card(M) \geqq \card(G)$ . Because $\card(f(S)) = \card(S)$ , we have $\card(M\!\smallsetminus\!S) \geqq \card(G\!\smallsetminus\!f(S))$ , and therefore there exists an injection $$\psi\!:G\!\smallsetminus\!f(S) \to M\!\smallsetminus\!S$$ (provided that $G\!\smallsetminus\!f(S) \neq \varnothing$ ; otherwise the mapping $f\!:S \to G$ would be a bijection). Define $$H \;:=\; S\cup\psi(G\!\smallsetminus\!f(S)),$$
Then apparently, $\varphi\!:H \to G$ is a bijection and $\varphi|S = f$ . Moreover, define the binary operation ``$*$ '' of the set $H$ by
We see first that
Secondly, $$h\ast\varphi^{-1}(e) \;=\; \varphi^{-1}\!\left(\varphi(h)\!\cdot\!\varphi(\varphi^{-1}(e))\right) \;=\; \varphi^{-1}(\varphi(h)) \;=\; h,$$ whence $\varphi^{-1}(e)$ is the right identity element of $H$ . Then, $$h\ast \varphi^{-1}\!\left((\varphi(h))^{-1}\right) \;=\; \varphi^{-1}\left(\varphi(h)\!\cdot\!\varphi\left(\varphi^{-1}(\varphi(h)^{-1})\right)\right) \;=\; \varphi^{-1}(e),$$ and accordingly $\displaystyle\varphi^{-1}\!\left((\varphi(h))^{-1}\right)$ is the right inverse of $h$ in $H$ . Consequently, $(H,\,\ast)$ is a group. The equation (1) implies that $$\varphi(h_1\ast h_2) \;=\; \varphi(h_1)\!\cdot\!\varphi(h_2),$$ whence $\varphi$ is an isomorphism from $H$ onto $G$ . Q.E.D.
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"injection can be extended to isomorphism" is owned by pahio.
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Cross-references: onto, isomorphism, implies, equation, right inverse, right identity, binary operation, bijection, mapping, proof, group isomorphism, group, injection, theorem
This is version 7 of injection can be extended to isomorphism, born on 2009-05-27, modified 2009-10-29.
Object id is 11804, canonical name is InjectionCanBeExtendedToIsomorphism.
Accessed 504 times total.
Classification:
| AMS MSC: | 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory ) | | | 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties) |
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Pending Errata and Addenda
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