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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.