The Kernel of a Group Homomorphism – Abstract Algebra The…

The Kernel of a Group Homomorphism – Abstract Algebra 

The kernel of a group homomorphism measures how far off it is from being one-to-one (an injection). Suppose you have a group homomorphism f:G → H. The kernel is the set of all elements in G which map to the identity element in H. It is a subgroup in G and it depends on f. Different homomorphisms between G and H can give different kernels.

If f is an isomorphism, then the kernel will simply be the identity element.

You can also define a kernel for a homomorphism between other objects in abstract algebra: rings, fields, vector spaces, modules. We will cover these in separate videos.

By: Socratica.
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The Kernel of a Group Homomorphism – Abstract Algebra The…

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