Let G and H be groups with the group operation in G denoted by juxtaposition and the group operation in H denoted by *. A homomorphism from G to H is a function f:G→H such that f(xy)=f(x)*f(y). The set of elements of G that map into the identity of H is called the kernel of the homomorphism and is denoted as Ker f. The kernel of a homomorphis is a normal subgroup of G.

An isomorphism is a homomorphism with an inverse; i.e., a one-to-one correspondence that preserves the group relation ships. If f is an isomorphism f:G→H then if the group operation in G is denoted by juxtaposition and the group operation in H is denoted by * then

for all x and y in G f(xy) = f(x)*f(y)
and
for all z and w in H f^-1(z*w) = f^-1(z)f^-1(w)

The image of the elements of G under a homomorphism f:G→H is a group and is denoted as Im f.

• First Isomorphism Theorem: If f:G→H is an isomorphism then there is an isomorphism between the factor group of G with respect to its normal subgroup Ker f (G/(Ker f)) and the image of G under f in H, im f.

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