Mathematics · Advanced university
Homomorphisms, kernels and quotient structure · revision notes
A structure-preserving map respects the relevant operation.
Key ideas and conditions
Homomorphism
A structure-preserving map respects the relevant operation.
Kernel
Elements mapped to the identity form the kernel; for a group homomorphism it is a normal subgroup.
Quotient
Cosets identify points differing by a kernel or a specified normal subgroup.
First isomorphism theorem
The quotient by the kernel is isomorphic to the image, with the induced map well-defined.
Explain without looking
- Show that a linear map's image is a subspace.
- V has dimension 11 and W is a 4-dimensional subspace. Find dim(V/W).
Further reading and review references
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