Like isomorphism, but does not have to be one-to-one: multiple different inputs can have the same output.
This brings us to the key intuition about group homomorphisms: they are a way to split out a larger group into smaller groups that retains a subset of the original structure.
As shown by the fundamental theorem on homomorphisms, each group homomorphism is fully characterized by a normal subgroup of the domain.
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A group homomorphism is a map preserving the group operation: . It also preserves the identity element and inverse elements. In an additive group, the same requirement reads . A group homomorphism preserves a way of combining inputs, even when several inputs have the same output.
Reduction modulo three from to is a group homomorphism: reducing a sum agrees with adding the reduced residues. For example, modulo six, and both sides reduce to one modulo three; the outputs and also add to one modulo three. This map is surjective but not injective, since zero and three both map to zero.
The sign of a permutation gives another example, from the symmetric group to the multiplicative group . A product of two odd permutations is even, matching . Its group kernel is , the alternating group of even permutations. This shows how a group homomorphism can retain a useful feature while forgetting other details.
