Permutation Groups [HB 18]

Changes:

• The semantics commands Verify and RandomSchreier now agree with the same named commands for matrix groups. That is, RandomSchreier(G) asserts the group order to be whatever the random schreier run ends with. On the other hand, the Verify command ignores any knowledge of the groups BSGS being complete and runs a verification algorithm. WARNING: RandomSchreier may assert an incorrect group order, and, if this happens, any future calculations with this group can go badly wrong. (There may be no sign of problems, there may be a crash, or an infinite loop for example.)

New Features:

• Magma's implementation of the Brownie-Cannon-Sims Verify routine has been upgraded to make use of block systems in the basic orbits and knowledge of a known base for the BSGS being verified. This has greatly improved both Verify and FPGroupStrong (which is based on this algorithm).

• Construction of conjugacy classes has been improved by revising the extension algorithm for lifting conjugacy classes through the soluble radical. It now uses less memory and runs faster.

• A version of Greg Butler's ``Inductive'' algorithm for computing conjugacy classes is now installed. This may be invoked using the intrinsic ClassesInductive, or by setting parameters Al or TFAl to "Inductive" in the standard Classes intrinsic. (The inductive algorithm is not yet part of the default strategy.)

• A new algorithm for computing DoubleCosetRepresentatives has been implemented. This refines double cosets along a subgroup chain using orbit-stabilizer methods. The intrinsic SubgroupChain gives access to the method used to construct the subgroup chain.

• Subgroup conjugacy and normalizer calculations now use a pre-processing phase where there is a reduction by the orbit structure of the subgroups.

• A function has been provided to determine whether a permutation group of moderate degree can be written as a permutational wreath product IsWreathProduct.