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Morfisme şi izomorfisme de grupuri

Fie ( G[pic], [pic]) şi ( G[pic], [pic]) două grupuri.
Definiţii:
Funcţia f : G[pic]→G[pic] se numeşte morfism ( omomorfism ) de grupuri dacă: f ( x[pic]y ) = f ( x ) [pic] f( y ), [pic] x [pic] G[pic].

Funcţia f : G[pic]→G[pic] se numeşte izomorfism de grupuri dacă: 1. f este morfism de grupuri; 2. f este funcţie bijectivă

Grupurile ( G[pic], [pic]) şi ( G[pic], [pic]) se numesc grupuri izomorfe şi se scrie G[pic]~ G[pic], dacă între ele există cel puţin un izomorfism de grupuri.

Definiţii: Un morfism f : G → G, se numeşte endomorfism al grupului G. Un izomorfism f : G → G, se numeşte automorfism al grupului G. Mulţimea endomorfismelor unui grup se notează End(G), iar mulţimea automorfismelor lui G se notează Aut(G).

Teoremă: Fie (G[pic], [pic]) şi ( G[pic], [pic]) două grupuri cu elementele neutre e[pic]şi e[pic], şi f : G[pic]→G[pic], un morfism de grupuri. Atunci : a) f ( e[pic])= f ( e[pic]) b) f ( x[pic]) = ( f (x) ) [pic], [pic]x [pic]G[pic] c) f ( x [pic])= ( f(x)) [pic] , [pic] x [pic]G[pic] şi n[pic]Z.

OBSERVAŢIE În scriere aditivă, relaţiile anterioare se scriu: f ( 0)= 0 f (-x)= - f(x), [pic] x [pic] G[pic] f ( nx)= nf(x), [pic]x [pic]G[pic] şi n [pic] Z
Teoremă:
Fie grupurile ( G[pic],·), ( G[pic],·) şi (G[pic],·). Dacă f : G[pic]→G[pic] şi g : G[pic]→G[pic] sunt morfisme de grupuri, atunci h : G[pic]→G[pic], h = g ◦ f este morfism de grupuri. Dacă f : G[pic]→G[pic]este izomorfism de grupuri, atunci [pic]: G[pic]→G[pic]este izomorfism de grupuri.
Demonstraţie
Avem succesiv: h ( xy) = g ( f ( xy )) = g ( f ( x ) · f ( y )) = g ( f ( x )) · g ( f ( y )) = h ( x ) · h ( y), [pic]x , y[pic]G[pic].
Funcţia fֿ[pic]: G[pic]→G[pic] este bijectivă.
Fie y [pic] şi y [pic] [pic] G [pic]. Deoarece f este bijectivă, rezultă că există x[pic]şi x [pic] [pic] G [pic], astfel ca f ( x[pic]) = y[pic]şi f ( x[pic]) = y [pic].
Avem:
f [pic]( y[pic]· y[pic])= f [pic] ( f (x[pic]) · f ( x[pic]))= f [pic] ( f ( x[pic]· x[pic])) = x[pic]· x[pic]
= f [pic]( y[pic]) · f [pic]( y[pic] )
Deci, f [pic] este izomorfism de grupuri.
q.e.d.
Teoremă: Fie (G, [pic] ) un grup. Atunci: a) ( End(G), [pic] ) este monoid; b) Aut( G ), [pic]) este grup.

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