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Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Use conjugation to test normality before defining a quotient group.
- Justify the conclusion "These left cosets do not form a quotient group with the usual product rule" using the stated assumptions.
Before you start
Permutation groups, cosets, and conjugation.
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
Can S₃/H be a quotient group when H={e,(12)}?
Why this math matters
Use conjugation to test normality before defining a quotient group. 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:
- Permutation products are composed right to left.
- The proposed quotient uses the usual coset multiplication.
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.
Check whether cosets support a group operation
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Can S₃/H be a quotient group when H={e,(12)}?
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
Take g=(123); g(12)g⁻¹=(23)
Conjugation relabels the two entries of the transposition.
Work through the mathematics
(23)∉H, so gHg⁻¹≠H
H is not invariant under conjugation and is therefore not normal.
Check the conclusion
These left cosets do not form a quotient group with the usual product rule
Multiplication of coset representatives would not be well-defined.
The result
These left cosets do not form a quotient group with the usual product rule
Multiplication of coset representatives would not be well-defined.
Common mistakes to catch
- Every subgroup has cosets, but not every subgroup gives a quotient group.
- Equal subgroup sizes do not establish normality.
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
Is A₃ normal in S₃?
Show a hint
Conjugation preserves permutation parity.
Reveal answer and explanation
Yes
Even permutations remain even under conjugation.
Practice 2
What is S₃/A₃ isomorphic to?
Show a hint
Its order is two.
Reveal answer and explanation
C₂
The two cosets record even and odd parity.
Take the idea with you
Identify the invariant information retained when permutations are reduced to parity.
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: Constrain subgroups using Sylow counting
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