Mathematics · lesson 2 of 3

Homomorphisms, kernels and quotient structure

A structure-preserving map respects the relevant operation.

Advanced university8 practice questionsNotes + worked examples

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Prerequisites: Groups, linear maps and equivalence classes.

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

Example 1

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.

Example 2

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

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

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

Show answer and explanation

Rank-nullity gives 8−3=5.

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

Sample answer and marking guidance

The multiples of 5, namely 5Z.

Credit exactly those integers mapped to the zero residue class.

3 A group homomorphism is injective exactly when its kernel is what?

A. EmptyB. The whole codomainC. The identity subgroupD. A set with two elements

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The identity subgroup

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

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

Show answer and explanation

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

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

Sample answer and marking guidance

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

Credit well-defined multiplication and representative independence.

6 What object is isomorphic to G/ker φ?

A. The set of all subgroupsB. The image of φC. Only ker φ itselfD. Every codomain of φ automatically

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The image of φ

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

7 Show that a linear map's image is a subspace.

Sample answer and marking guidance

Zero is T(0), and aT(u)+bT(v)=T(au+bv) is again in the image.

Credit zero and closure.

8 V has dimension 11 and W is a 4-dimensional subspace. Find dim(V/W).

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The quotient retains 11−4 independent directions.

Learn the ideas

Homomorphism

Kernel

Quotient

First isomorphism theorem

Worked examples

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

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

Check your understanding

Further reading and review references