Regular Permutation Groups Library

The Hopf-Galois structures on a Galois extension \(L/K\) with \(G=Gal(L/K)\) are in 1-1 correspondence with the regular subgroups \(N\leq B=Perm(G)\) which are normalized by the left regular representation \(\lambda(G)\leq B=Perm(G)\).

Regularity implies that \(|G|=|N|\) but other than that, \(G\) and \(N\) need not be isomorphic.

As such, one can tabulate \(R(G,[M])\) which are the set of all such regular \(N\) where \(N\cong M\) as \(G\) and \([M]\) range over all the possible isomorphism classes of groups of a given order \(n\).

As regularity can be defined for subgroups of \(S_n\), just as for \(Perm(G)\), we can pick a representative set of regular subgroups \(\{G_i\}\) embedded in \(S_n\) and for these, compute \(R(G_i,[G_j])\) over all possible pairs of isomorphism classes of groups of order \(n\).

We have computed a number of collections of \(R(G_i,[G_j])\) in GAP, using the 'reflection principle' relating \(R(G_i,[G_j]\) and the sets \(S(G_j,[G_i])\) where the latter are the regular subgroups of \(Hol(G_j)=Norm_{S_n}(G_j)\) which are isomorphic to \(G_i\).

How to use these libraries.

After loading a particular library in GAP, for example

Read("R6.g");

One must also Read the 'MakeLib.g' file

Read("MakeLib.g");

(for each Rn.g that one loads) to actually compute \(R(G_i,[G_j])\) as the Rn.g file contains the elements in \(B=S_n\) which conjugate each \(G_i\) to the given \(N\in R(G_j,[G_i]\).

What this yields are the following objects

Downloads

MakeLib.g

Note, some of these are rather large.