Notation for Database of Simple Groups

In all cases the data input for the permutation version and corresponding finitely presented version of a group agree in the following sense. Substituting the generating permutations of the permutation version into the words given in the finitely presented version for conjugacy class representatives and subgroup generators will always yield the permutations given in the permutation version.

Conjugacy Classes

In the ClassInfo listing, the classes that contain elements of order n are named nA, nB, nC, etc. Class names of the form Y*k, Y**k, Y**, Y* give the additional information that the class can be obtained from the most recently named class nY by applying the algebraic conjugacy operator of k th powers, -k th powers, inverses, other powers respectively. For each class the order of the centralizer in G of a typical element x of the class is given. A sequence of class representatives for either the permutation representation or the finitely presented group may be obtained from the simple group record X. A representative x for a particular class may be obtained by typing the statements:

SimRecordRequire(~X, "Reps");
x := X`Reps[SimClassNameIndex(X, "nY")];

Maximal Subgroups

The information regarding the structure of a subgroup is given in a notation similar to the Atlas.

• m denotes a cyclic group of order m
• m^n denotes a direct product of n cyclic groups of order m
• A x B is the direct product of groups A and B
• A.B or AB is a group with normal subgroup A and quotient B
• A:B is a case of A.B where the extension splits
• A'B is a case of A.B where the extension does not split
• p^(m+n) is a case of (p^m).(p^n)

The specification of H locates a copy of H inside G. For example N(2A) is the normalizer of an involution in 2A; N(5AB) is the normalizer of a group of order 5 containing elements of classes 5A and 5B; N(3^3)=N(3AB4 C3 D6) is the normalizer of an elementary abelian group of order 27 whose 13 cyclic subgroups number 4 containing both classes 3A and 3B, 3 containing 3C only, 6 containing 3D only; N(2A,2C,3A,3B,...) is the normalizer of a group containing elements in the indicated classes among others. Within Magma, the maximal subgroups relative to the finitely presented group are stored in the sequence MaxF, while the maximal subgroups relative to the permutation group are stored in the sequence Max.

Sylow Subgroups

If the prime p divides |G| then generators for a Sylow p-subgroup are input to Magma provided that the Sylow subgroup is not cyclic. Hence if X is the simple group record for the group PSU42, say, a Sylow 2-subgroup relative to the finitely presented group is stored (on request) as X`SylF[2] while a Sylow 2-subgroup relative to the permutation group is stored as X`Syl[2].