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# subgroup-ladders
This repository contains a gap implementation of an algorithm described by Bernd Schmalz [1, Theorem 3.1.1] to compute subgroup ladders of permutation groups.
Solutions of some problems in group theory can relatively easy be transferred to a sub- or supergroup if the index is small.
Let G be a permutation group on the set {1,…,n}.
So one might try to find a series of subgroups of subgroups G = H_0,…,H_k = S_n of the symmetric group S_n such that H_{i-1} is a subgroup of H_i for every i and transfer the solution of a problem for the symmetric group step by step to G.
Sometimes it is not possible to find such a series with small indices between consecutive subgroups.
This is where subgroup ladders may make sense:
A subgroup ladder is series of subgroups G = H_0,…,H_k = S_n of the symmetric group such that for every 1≤i≤k, H_i is a subgroup of H_{i-1} or H_{i-1} is a subgroup of H_i.
So we sometimes go up to a larger group in order to keep the indices small.
If G is a Young subgroup of S_n, the algorithm in this repository can find a subgroup ladder of G such that the indices are at most the degree of the permutation group.
```text
S_n
|
|
|
H_1
| H_3
| /|
| / |
| / |
H_2 |
H_4
| H_6
| / |
| / |
| / |
H_5 |
|
H_7 = G
```
## References
[1] B. Schmalz. Verwendung von Untergruppenleitern zur Bestimmung von Doppelnebenklassen. Bayreuther Mathematische Schriften, 31, S.109--143, 1990.
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