# Homomorphisms, kernels and quotient structure

> A structure-preserving map respects the relevant operation.

Course: [Abstract algebra and discrete proofs](https://studybrix.com/en/courses/algebra-and-discrete-proofs/index.md)
Level: advanced
Prerequisites: Groups, linear maps and equivalence classes.

## 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.

## Practice questions

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

Answer: 5

Explanation: Rank-nullity gives 8−3=5.

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

Sample answer / marking guidance: The multiples of 5, namely 5Z.

Explanation: Credit exactly those integers mapped to the zero residue class.

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

A. Empty
B. The whole codomain
C. The identity subgroup
D. A set with two elements

Answer: The identity subgroup

Explanation: 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?

Answer: 4

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 / marking guidance: Normality ensures the product of cosets is independent of the representative choices.

Explanation: Credit well-defined multiplication and representative independence.

### 6. What object is isomorphic to G/ker φ?

A. The set of all subgroups
B. The image of φ
C. Only ker φ itself
D. Every codomain of φ automatically

Answer: The image of φ

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

Explanation: Credit zero and closure.

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

Answer: 7

Explanation: The quotient retains 11−4 independent directions.

## Further reading

- https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/
- https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.