Calculator Input
This calculator is designed for finite abelian groups written as products of cyclic factors, such as C12 × C18 × C20.
Example Data Table
These examples illustrate how the calculator interprets finite abelian groups and counts prime-order composition factors.
| Input cyclic factors | Group notation | Group order | Composition length | One possible factor multiset |
|---|---|---|---|---|
| 12, 18 | C12 × C18 | 216 | 6 | C2, C2, C2, C3, C3, C3 |
| 8, 9 | C8 × C9 | 72 | 5 | C2, C2, C2, C3, C3 |
| 4, 6, 10 | C4 × C6 × C10 | 240 | 6 | C2, C2, C2, C2, C3, C5 |
| 27 | C27 | 27 | 3 | C3, C3, C3 |
Formula Used
This calculator uses the finite abelian group decomposition entered through cyclic factors.
Each input factor ni is factorized into primes. If ni = ∏ pa, then the total number of prime factors with multiplicity contributes to the composition length.
Here, Ω(n) means the total number of prime factors of n, counting repeated primes.
The subgroup order drops by one prime factor at every step of the chain.
Every quotient Gr-1 / Gr is a simple cyclic group of prime order, written as Cp.
How to Use This Calculator
- Enter the cyclic factors of your finite abelian group, such as 12, 18, 20.
- Choose a reduction strategy. It changes the displayed chain order only.
- Select the display sorting option when you want subgroup factors shown in increasing order.
- Press Find Composition Series.
- Read the summary cards for order, length, and factor counts.
- Inspect the composition factor badges and the subgroup chain table.
- Use the Plotly graph to follow subgroup order through each prime reduction.
- Export the result using the CSV or PDF buttons.
FAQs
1) What kind of groups does this calculator support?
It supports finite abelian groups entered as products of cyclic factors, such as C12 × C18 × C20. It does not attempt general nonabelian composition-series computation.
2) Why are the composition factors always prime-order cyclic groups?
For finite abelian groups, simple quotient groups must be cyclic of prime order. That is why every displayed composition factor has the form Cp.
3) What does the composition length mean?
It is the number of simple factors in the composition series. In this calculator, it equals the total count of prime factors, with multiplicity, across all input cyclic orders.
4) Why can different strategies produce different chains?
Different strategies remove prime divisors in different orders. The actual chain may vary, but the multiset of composition factors remains the same by the Jordan–Hölder theorem.
5) Does sorting subgroup factors change the mathematics?
No. Sorting affects display only. It simply rewrites the direct-product factors in a cleaner order for reading, exporting, and comparing results.
6) Can I enter repeated cyclic factors?
Yes. Repeated values are fully valid. For example, entering 4, 4, 9 represents C4 × C4 × C9, and the calculator handles each component separately.
7) What does the graph show?
The Plotly graph shows the subgroup order after each reduction step. It helps you visualize how the chain descends from the whole group to the trivial subgroup.
8) Why is this useful for teaching or revision?
It connects decomposition, subgroup chains, quotient simplicity, and prime factorization in one place. That makes examples faster to verify and easier to explain.