Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Compute an image and kernel for reduction modulo four.
- Justify the conclusion "Z/ker φ≅Z/4Z" using the stated assumptions.
Before you start
Groups under addition and modular arithmetic.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Analyze φ:Z→Z/4Z given by φ(n)=[n].
Why this math matters
Compute an image and kernel for reduction modulo four. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Both group operations are addition.
- The codomain consists of residue classes, not ordinary integers.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Find the information lost by a group homomorphism
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Analyze φ:Z→Z/4Z given by φ(n)=[n].
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
φ(a+b)=[a+b]=[a]+[b]
Reduction respects the group operation, so φ is a homomorphism.
Work through the mathematics
ker φ=4Z; im φ=Z/4Z
Multiples of four map to the identity, and every residue class is reached.
Check the conclusion
Z/ker φ≅Z/4Z
Collapsing exactly the invisible differences recovers the image, illustrating the first isomorphism theorem.
The result
Z/ker φ≅Z/4Z
Collapsing exactly the invisible differences recovers the image, illustrating the first isomorphism theorem.
Common mistakes to catch
- The kernel is the preimage of the identity, not of an arbitrary value.
- A surjective map need not be injective.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is the kernel of reduction modulo seven?
Show a hint
Find integers mapping to [0].
Reveal answer and explanation
7Z
Exactly the multiples of seven have zero remainder.
Practice 2
Is n↦n+1 a homomorphism Z→Z under addition?
Show a hint
Test preservation of zero or addition.
Reveal answer and explanation
No
It sends the identity zero to one, and adds an extra one on each side inconsistently.
Take the idea with you
Interpret a checksum as retaining some information while identifying many original inputs.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Separate a polynomial quotient from its remainder
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.