injection can be extended to isomorphism

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 isomorphismφ:HG  such that  φ|S=f.

Proof.  Let M be a set such that  card(M)card(G).  Because  card(f(S))=card(S), we have  card(MS)card(Gf(S)),  and therefore there exists an injection


(provided that  Gf(S);  otherwise the mappingf:SG would be a bijection).  Define

φ(h):={f(h)  forhS,ψ-1(h)forhHS.

Then apparently,  φ:HG  is a bijection and  φ|S=f.  Moreover, define the binary operationMathworldPlanetmath*” of the set H by

h1h2:=φ-1(φ(h1)φ(h2)). (1)

We see first that

(h1h2)h3 =φ-1(φ(φ-1(φ(h1)φ(h2)))φ(h3))



whence φ-1(e) is the right identityPlanetmathPlanetmath element of H.  Then,


and accordingly φ-1((φ(h))-1) is the right inverse of h in H.  Consequently, (H,) is a group.  The equation (1) implies that


whence φ is an isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from H onto G.  Q.E.D.

Title injection can be extended to isomorphism
Canonical name InjectionCanBeExtendedToIsomorphism
Date of creation 2013-03-22 18:56:50
Last modified on 2013-03-22 18:56:50
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 20A05
Classification msc 03E20
Related topic RestrictionPlanetmathPlanetmathPlanetmathPlanetmath
Related topic Cardinality