Definition·D045
Group homomorphism
A function between groups that turns products into products: it preserves the operation.
In words
For any a and b, if both lie in G then phi sends their product to the product of their images.
Rests onA02
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
This single condition already forces
to preserve the rest of the structure: it sends the identity to the identity and inverses to inverses (L37). A bijective homomorphism is an isomorphism, under which the two groups are indistinguishable. The elements
crushes to the identity form its kernel, a normal subgroup measuring the failure of injectivity, while the image is a subgroup of
. Once the integers and modular arithmetic are built, reduction
is the basic example.