Definition·D047
Kernel of a homomorphism
The elements a homomorphism sends to the identity of the target group.
The kernel of a homomorphism
is the set of elements it sends to the identity of
:
In words
The kernel of φ is the set of elements it sends to the identity of H.
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
The kernel is a normal subgroup of
(L38), and
is injective exactly when its kernel is trivial,
(L38): the kernel is the precise obstruction to injectivity. Kernels turn out to be exactly the normal subgroups, each being the kernel of its quotient map, once quotient groups are built. Together with the image, the kernel records the two ways a homomorphism can fail to be an isomorphism.