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

2-2018 Teaching by misleading

2 months ago

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