SOTGrps

This package for the GAP computer algebra system is complementary to an MPhil thesis “Groups of small order type” and the joint paper “Groups whose order factorise into at most four primes” (Dietrich, Eick, & Pan, 2020) available at https://doi.org/10.1016/j.jsc.2021.04.005.

Using this package requires GAP, you can get it from https://www.gap-system.org/Download/. If you have GAP installed, then please unzip the file into the pkg folder in GAP, and then simply run the command LoadPackage("sotgrps") in GAP.

The main user functions are given in the file SOTGrps.gi.

User functions:

• NumberOfSOTGroups(n): returns the number of isomorphism types of groups of order n.

• AllSOTGroups(n): takes in a number n that factorises into at most 4 primes or of the form p^4q (p, q are distinct primes), and outputs a complete and duplicate-free list of isomorphism class representatives of the groups of order n. If a group is solvable, then it constructs the group using refined polycyclic presentations; otherwise the group is given as a permutation group.

• SOTGroup(n, i): returns the i-th group with respect to the ordering of the list AllSOTGroups(n) without constructing all groups in the list.

• IdSOTGroup(G): returns false if G is not a group or |G| is not available; otherwise returns the SOT-group ID (n, i), where n = |G| and G is isomorphic to SOTGroup(n, i).

• IsSOTAvailable(n): returns true if the groups of order n are available.

• SOTGroupsInformation(n): returns a brief comment on the enumeration of the isomorphism types of groups of order n.

Note that the construction of small groups could be different to the existing library, for which reason the list of groups for a given order may not have the same order as enumerated by the IdGroup / IdSmallGroup function.

In particular, with SOTGroup(n, i), we construct the i-th group of order n in our SOT-group list.

IdSOTGroup(group) returns the group ID in line with SOTGroup(n, i), namely, the position of the input group of order n in the list constructed by AllSOTGroups(n).

References

[1] X. Pan, Groups of small order type. MPhil thesis at Monash University. https://xpan-eileen.github.io/documents/Thesis_Groups_of_small_order_type.pdf

[2] H. Dietrich, B. Eick, & X. Pan, Groups whose order factorise into at most four primes. In: Journal of Symbolic Computation (108) (2022), pp. 23–40. https://doi.org/10.1016/j.jsc.2021.04.005

License

The SOTGrps package is free software; you can redistribute and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your opinion) any later version.

The SOTGrps package is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.