Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Use disjoint equal-size cosets to constrain subgroup orders.
- Justify the conclusion "[S₃:H]=6/2=3" using the stated assumptions.
Before you start
Groups, subgroups, and permutations.
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
For H={e,(12)} inside S₃, how many left cosets are there?
Why this math matters
Use disjoint equal-size cosets to constrain subgroup orders. 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:
- The group is finite.
- H contains the identity and the specified transposition.
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.
Count subgroup cosets in a finite symmetry group
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For H={e,(12)} inside S₃, how many left cosets are there?
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
|S₃|=6 and |H|=2
A permutation of three labels has six possible arrangements.
Work through the mathematics
Left multiplication is a bijection H→gH
Every coset therefore contains two elements.
Check the conclusion
[S₃:H]=6/2=3
Cosets partition the group, giving Lagrange's counting relation.
The result
[S₃:H]=6/2=3
Cosets partition the group, giving Lagrange's counting relation.
Common mistakes to catch
- A subgroup of S₃ need not be normal.
- Divisibility is necessary but is not a general existence theorem.
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
Can a group of order ten have a subgroup of order four?
Show a hint
Subgroup order must divide group order.
Reveal answer and explanation
No
Four does not divide ten.
Practice 2
Does every divisor guarantee a subgroup in every finite group?
Show a hint
Lagrange gives a necessary condition.
Reveal answer and explanation
No
The counting theorem alone does not provide a converse.
Take the idea with you
Use an orbit of equivalent configurations to explain why cosets have equal size.
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: Find the information lost by a group homomorphism
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