Kernel (algebra)

The kernel of a group homomorphism from "G" to "H" is the subset of the elements from "G" that arrive to the Identity element of "H". 
Mathematically: formula_1. Since a group homomorphism preserves identity elements, the identity element of "G" must belong to the kernel subset.
Property.
The homomorphism is injective if and only if its kernel is only the identity elements of "G".
Proof: If "f" were not injective, then the non-injective elements can form a distinct element of its kernel: there would exist formula_2 such that formula_3 and formula_4. Thus formula_5. "f" is a group homomorphism, so inverses and group operations are preserved, giving formula_6; in other words, formula_7, and ker "f" would not be the singleton. Conversely, distinct elements of the kernel violate injectivity directly: if there would exist an element formula_8, then formula_9, thus "f" would not be injective.