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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: An for n
999,
L2(q), L3(q) and L4(q) for all q, L5(p), S4(p) and U3(p)
for all primes p, Ld(2) for d
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,...Sn]
where S1 is the soluble radical of G, Sn = 1, and each quotient
Si/Si+1 is elementary abelian, which is canonical in the sense that
any group isomorphism G
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.
Next: Permutation Groups [HB 18]
Up: Groups
Previous: Groups