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of , then the kernel of is the preimage of the singleton set ; that is, the subset of consisting of all those elements of that are mapped by to the element .

The kernel is usually denoted (or a variation). In symbols:
:

Since a group homomorphism preserves identity elements, the identity element of must belong to the kernel. The homomorphism is injective if and only if its kernel is only the singleton set .

is a subgroup of and further it is a normal subgroup.

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