Kernel (algebra)

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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: . Since a group homomorphism preserves identity elements, the identity element of G must belong to the kernel subset.

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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 such that and . Thus . f is a group homomorphism, so inverses and group operations are preserved, giving ; in other words, , and ker f would not be the singleton. Conversely, distinct elements of the kernel violate injectivity directly: if there would exist an element , then , thus f would not be injective.

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