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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.
Graduate chapters and video availability01 · Read and understand
What you will learn
- Combine divisibility and congruence conditions on Sylow subgroup counts.
- Justify the conclusion "n₇=1, hence the Sylow 7-subgroup is normal" using the stated assumptions.
Before you start
Finite groups and prime factorization.
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 a group G of order 21, what can be said about its Sylow 7-subgroup?
Why this math matters
Combine divisibility and congruence conditions on Sylow subgroup counts. 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:
- G is a finite group of order exactly 21.
- Sylow's theorems are available as established results.
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.
Constrain subgroups using Sylow counting
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For a group G of order 21, what can be said about its Sylow 7-subgroup?
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
21=3·7, so a Sylow 7-subgroup has order seven
The largest power of seven dividing the group order determines its size.
Work through the mathematics
n₇ divides 3 and n₇≡1 mod 7
Sylow's counting restrictions leave candidates one and three before the congruence test.
Check the conclusion
n₇=1, hence the Sylow 7-subgroup is normal
Conjugation permutes Sylow subgroups; a unique one must be fixed.
The result
n₇=1, hence the Sylow 7-subgroup is normal
Conjugation permutes Sylow subgroups; a unique one must be fixed.
Common mistakes to catch
- Do not confuse subgroup order with the number of such subgroups.
- A necessary counting restriction is not always a complete classification.
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 are the possibilities for n₃ in order 21?
Show a hint
It divides seven and is one modulo three.
Reveal answer and explanation
1 or 7
Both divisors satisfy the congruence.
Practice 2
Does n₇=1 prove G is cyclic?
Show a hint
Normality alone is weaker than commutativity.
Reveal answer and explanation
No
Additional structure is needed; groups of order 21 need not be cyclic.
Take the idea with you
Use Sylow counts to narrow possible symmetry structures before constructing examples.
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: Use an ideal to make quotient multiplication consistent
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