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 a permutation product and recognize noncommutativity.
- Justify the conclusion "ab≠ba" using the stated assumptions.
Before you start
Functions and cycle notation.
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
In S₃, compose a=(12) and b=(23), applying the rightmost permutation first.
Why this math matters
Compute a permutation product and recognize noncommutativity. 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:
- Products are read right to left.
- The permutations act on the set {1,2,3}.
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.
Track composition order in permutations
PausedQuestion: Start with the question. Paused.
Question
Start with the question
In S₃, compose a=(12) and b=(23), applying the rightmost permutation first.
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
ab: 1→2, 2→3, 3→1
Follow b first and then a for each label.
Work through the mathematics
ab=(123); ba=(132)
Reversing the operation order reverses this three-cycle.
Check the conclusion
ab≠ba
Permutation composition is associative but need not commute.
The result
ab≠ba
Permutation composition is associative but need not commute.
Common mistakes to catch
- State the composition convention before calculating.
- Associativity does not imply commutativity.
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 (123)²?
Show a hint
Follow each label twice.
Reveal answer and explanation
(132)
The successive images are 1→3, 3→2, 2→1.
Practice 2
What is the inverse of (12)?
Show a hint
Apply the swap twice.
Reveal answer and explanation
(12)
Two identical swaps return every label to its starting place.
Take the idea with you
Model the order sensitivity of swapping positions in a three-item display.
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: Count subgroup cosets in a finite symmetry group
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