47.7 Finite Perfect Groups

returns a group which is isomorphic to the library group specified by the size number [ size, n ] or by the two separate arguments size and n, assuming a default value of n = 1. The optional argument filt defines the filter in which the group is returned. Possible filters so far are IsPermGroup and IsSubgroupFpGroup. In the latter case, the generators and relators used coincide with those given in HP89.

gap> G := PerfectGroup(IsPermGroup,6048,1);
U3(3)

As all groups are stored by presentations, a permutation representation is obtained by coset enumeration. Note that some of the library groups do not have a faithful permutation representation of small degree. Computations in these groups may be rather time consuming.

gap> G:=PerfectGroup(IsPermGroup,129024,2);
L2(8) N ( 2^6 E 2^1 A ) C 2^1
gap> NrMovedPoints(G);
14336

This attribute is set for all groups obtained from the perfect groups library and has the value [size,nr] if the group is obtained with these parameters from the library.

returns the number of non-isomorphic perfect groups of size size for each positive integer size up to 10⁶ except for the eight sizes listed at the beginning of this section for which the number is not yet known. For these values as well as for any argument out of range it returns fail.

returns the number of perfect groups of size size which are available in the library of finite perfect groups. (The purpose of the function is to provide a simple way to formulate a loop over all library groups of a given size.)

SizeNumbersPerfectGroups returns a list of pairs, each entry consisting of a group order and the number of those groups in the library of perfect groups that contain the specified factors factor1, factor2, ... among their composition factors.

Each argument must either be the name of a simple group or an integer which stands for the product of the sizes of one or more cyclic factors. (In fact, the function replaces all integers among the arguments by their product.)

The following text strings are accepted as simple group names.

·

An or A(n) for the alternating groups A_n, 5 £ n £ 9, for example A5 or A(6).

·

Ln(q) or L(n,q) for PSL(n,q), where n Î {2,3} and q a prime power, ranging

   °

for n = 2 from 4 to 125

   °

for n = 3 from 2 to 5

·

Un(q) or U(n,q) for PSU(n,q), where n Î {3,4} and q a prime power, ranging

   °

for n = 3 from 3 to 5

   °

for n = 4 from 2 to 2

·

Sp4(4) or S(4,4) for the symplectic group S(4,4),

·

Sz(8) for the Suzuki group Sz(8),

·

Mn or M(n) for the Mathieu groups M₁₁, M₁₂, and M₂₂, and

·

Jn or J(n) for the Janko groups J₁ and J₂.

Note that, for most of the groups, the preceding list offers two different names in order to be consistent with the notation used in HP89 as well as with the notation used in the DisplayCompositionSeries command of GAP. However, as the names are compared as text strings, you are restricted to the above choice. Even expressions like L2(2^5) are not accepted.

As the use of the term PSU(n,q) is not unique in the literature, we mention that in this library it denotes the factor group of SU(n,q) by its centre, where SU(n,q) is the group of all n ×n unitary matrices with entries in GF(q²) and determinant 1.

The purpose of the function is to provide a simple way to formulate a loop over all library groups which contain certain composition factors.

DisplayInformationPerfectGroups displays some invariants of the n-th group of order size from the perfect groups library.

If no value of n has been specified, the invariants will be displayed for all groups of size size available in the library. The information provided for G includes the following items:

·

a headline containing the size number [ size, n ] of G in the form size.n (the suffix .n will be suppressed if, up to isomorphism, G is the only perfect group of order size),

·

a message if G is simple or quasisimple, i.e., if the factor group of G by its centre is simple,

·

the ``description'' of the structure of G as it is given by Holt and Plesken in HP89 (see below),

·

the size of the centre of G (suppressed, if G is simple),

·

the prime decomposition of the size of G,

·

orbit sizes for a faithful permutation representation of G which is provided by the library (see below),

·

a reference to each occurrence of G in the tables of section 5.3 of HP89. Each of these references consists of a class number and an internal number (i,j) under which G is listed in that class. For some groups, there is more than one reference because these groups belong to more than one of the classes in the book.

gap> DisplayInformationPerfectGroups( 30720, 3 );
#I Perfect group 30720:  A5 ( 2^4 E N 2^1 E 2^4 ) A
#I   size = 2^11*3*5  orbit size = 240
#I   Holt-Plesken class 1 (9,3)
gap> DisplayInformationPerfectGroups( 30720, 6 );
#I Perfect group 30720:  A5 ( 2^4 x 2^4 ) C N 2^1
#I   centre = 2  size = 2^11*3*5  orbit size = 384
#I   Holt-Plesken class 1 (9,6)
gap> DisplayInformationPerfectGroups( Factorial( 8 ) / 2 );
#I Perfect group 20160.1:  A5 x L3(2) 2^1
#I   centre = 2  size = 2^6*3^2*5*7  orbit sizes = 5 + 16
#I   Holt-Plesken class 31 (1,1) (occurs also in class 32)
#I Perfect group 20160.2:  A5 2^1 x L3(2)
#I   centre = 2  size = 2^6*3^2*5*7  orbit sizes = 7 + 24
#I   Holt-Plesken class 31 (1,2) (occurs also in class 32)
#I Perfect group 20160.3:  ( A5 x L3(2) ) 2^1
#I   centre = 2  size = 2^6*3^2*5*7  orbit size = 192
#I   Holt-Plesken class 31 (1,3)
#I Perfect group 20160.4:  simple group  A8
#I   size = 2^6*3^2*5*7  orbit size = 8
#I   Holt-Plesken class 26 (0,1)
#I Perfect group 20160.5:  simple group  L3(4)
#I   size = 2^6*3^2*5*7  orbit size = 21
#I   Holt-Plesken class 27 (0,1)

For any library group G, the library files do not only provide a presentation, but, in addition, a list of one or more subgroups S₁, ¼, S_r of G such that there is a faithful permutation representation of G of degree å_{i = 1}^r [G:S_i] on the set { S_i g \mid 1 £ i £ r,  g Î G } of the cosets of the S_i. This allows one to construct the groups as permutation groups. The DisplayInformationPerfectGroups function displays only the available degree. The message

orbit size = 8

in the above example means that the available permutation representation is transitive and of degree 8, whereas the message

orbit sizes = 5 + 16

The notation used in the ``description'' of a group is explained in section 5.1.2 of HP89. We quote the respective page from there:

advanceleftskip bymanindent advancerightskip bymanindent Within a class Q # p, an isomorphism type of groups will be denoted by an ordered pair of integers (r,n), where r ³ 0 and n > 0. More precisely, the isomorphism types in Q # p of order p^r \mid Q \mid will be denoted by (r,1), (r,2), (r,3), ¼ . Thus Q will always get the size number (0,1).

In addition to the symbol (r,n), the groups in Q  #  p will also be given a more descriptive name. The purpose of this is to provide a very rough idea of the structure of the group. The names are derived in the following manner. First of all, the isomorphism classes of irreducible F_pQ-modules M with \mid Q \mid \mid M \mid   £ 10⁶, where F_p is the field of order p, are assigned symbols. These will either be simply p^x, where x is the dimension of the module, or, if there is more than one isomorphism class of irreducible modules having the same dimension, they will be denoted by p^x, p^x¢, etc. The one-dimensional module with trivial Q-action will therefore be denoted by p¹. These symbols will be listed under the description of Q. The group name consists essentially of a list of the composition factors working from the top of the group downwards; hence it always starts with the name of Q itself. (This convention is the most convenient in our context, but it is different from that adopted in the ATLAS CCN85, for example, where composition factors are listed in the reverse order. For example, we denote a group isomorphic to SL(2,5) by A₅ 2¹ rather than 2 ·A₅.)

Some other symbols are used in the name, in order to give some idea of the relationship between these composition factors, and splitting properties. We shall now list these additional symbols.