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diff --git a/README.md b/README.md new file mode 100644 index 0000000..48e4066 --- /dev/null +++ b/README.md @@ -0,0 +1,39 @@ +# 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. |