Definition of homomorphism.
Proof that e is mapped to e by any homomorphism.
Proof that inverses are mapped to inverses by any homomorphism.
Definition of Image.
Definition of Kernel.
Properties of the Kernel.
- subgroup;
- normal subgroup.
Any normal subgroup is the kernel of a homomorphism.
Example homomorphism.
f: GL(n,R) -> R_x
f(A) = det(A)
f has as kernel the matrices with det=1, also called SL(n,R). ( Special linear group )
Example homomorphism.
f: Sn->GL(n,R)
f(p)=Ap ( permutation matrix associated with p )
f( (1,2,3) ) = {{0,0,1}, {1,0,0}, {0,1,0} }
Definition center of G.
Example homomorphism G-> Aut(G) i.e. Klein4 -> S3
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