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Mathematics · Advanced university

Homomorphisms, kernels and quotient structure

A structure-preserving map respects the relevant operation.

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Prerequisites

Groups, linear maps and equivalence classes.

Study notes

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.

Worked examples

A linear map from R^8 has a kernel of dimension 3. What is the dimension of its image?

Rank-nullity gives 8−3=5.

5

For φ:Z→Z/5Z given by reduction modulo 5, describe its kernel.

Credit exactly those integers mapped to the zero residue class.

The multiples of 5, namely 5Z.

Check your understanding

  1. A group homomorphism is injective exactly when its kernel is what?

    1. Empty
    2. The whole codomain
    3. The identity subgroup
    4. A set with two elements

  2. A finite group of order 24 has a homomorphism with kernel order 6. What is the image order?

  3. Why must a group subgroup be normal before the usual coset multiplication defines a quotient group?

  4. What object is isomorphic to G/ker φ?

    1. The set of all subgroups
    2. The image of φ
    3. Only ker φ itself
    4. Every codomain of φ automatically

Answers and marking guidance

Answers and marking notes
  1. The identity subgroup

    A trivial kernel means no two distinct domain elements have the same image.

  2. 4

    The first isomorphism theorem or equal fibers gives 24/6=4.

  3. Normality ensures the product of cosets is independent of the representative choices.

    Credit well-defined multiplication and representative independence.

  4. The image of φ

    Surjectivity onto the whole codomain is not automatic, so the theorem identifies the image.

Further reading and review references

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