Finite Groups [HB 17]

New Features:

• The IsIsomorphic intrinsic has been improved by Derek Holt. This may now be applied across three types of finite groups, to permutation, finite matrix and finite soluble groups.

• The stored data necessary for the insoluble composition factors that may be encountered in group structure algorithms has been extended. The list in the database is: All simple groups of order less than 1.6×10⁷, plus M₂₄, HS, J₃, McL, Sz(32) and L₆(2). There are also special routines to handle numerous other groups. These include: A_n for n$\le$999, L₂(q), L₃(q) and L₄(q) for all q, L₅(p), S₄(p) and U₃(p) for all primes p, L_d(2) for d$\le$14, U₆(2), S₈(2), O^±₈(2), O^±₁₀(2), S₆(3), O₇(3), G₂(4), G₂(5), ³D₄(2), ²F₄(2)', Co₂, Co₃, He, Fi₂₂.

• There is a new intrinsic ElementaryAbelianSeriesCanonical(G), where G is a permutation, finite matrix or finite soluble group. This produces a sequence [S₁, S₂,...S_n] where S₁ is the soluble radical of G, S_n = 1, and each quotient S_i/S_i+1 is elementary abelian, which is canonical in the sense that any group isomorphism G$\to$H will take ElementaryAbelianSeriesCanonical(G) to ElementaryAbelianSeriesCanonical(H).

• The handling of subgroup relationships now uses new methods which are much faster when there are many subgroups of a group to be dealt with.

• Black-box groups have been added to support the development of algorithms for black-box recognition of groups.

• Non-constructive recognition of simple groups using a Monte-Carlo algorithm is now available. The method was developed by G. Malle and E. O'Brien and incorporates the algorithm of Babai et al.

• The Kantor-Seress algorithm for black box recognition of linear groups has been installed.

• The Bratus-Pak algorithm for black box recognition of the alternating and symmetric groups has been implemented by Holt.

• The Babai et al algorithm for black-box recognition of the alternating and symmetric groups has been implemented by Roney-Dougal.

• Much of the information in the online Atlas of Simple Groups maintained by Robert Wilson at Birmingham is now available as a Magma database. In particular, the list of ordinary and modular representations is available.