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Results for other groups.


The methods described above have been developed with a particular interest in the tables of marks of simple and almost simple groups. While it should be possible to adapt them to larger classes of groups, it should be noted that arguments involving Sylow subgroups (in particular Corollary 4.8) loose their meaning when the table of marks of a p-group is to be determined.

Table 4 in the appendix lists the numbers of subgroups and the numbers of conjugacy classes of subgroups for the following groups: all projective special linear groups tex2html_wrap_inline4707 of order tex2html_wrap_inline4709 ; the projective special linear groups tex2html_wrap_inline4711 for n = 3, 4, 5; the alternating groups tex2html_wrap_inline4715 for tex2html_wrap_inline4717 ; the symmetric groups tex2html_wrap_inline4719 for tex2html_wrap_inline4721 ; the unitary groups tex2html_wrap_inline4723 for n = 3, 4, 5 and tex2html_wrap_inline4727 ; the Suzuki group Sz(8); the symplectic group tex2html_wrap_inline4731 ; the Mathieu groups tex2html_wrap_inline4733 , tex2html_wrap_inline4735 , tex2html_wrap_inline4737 and tex2html_wrap_inline4739 plus their automorphism groups tex2html_wrap_inline4741 and tex2html_wrap_inline4743 ; the Janko groups tex2html_wrap_inline4745 , tex2html_wrap_inline4747 and tex2html_wrap_inline4749 ; and the McLaughlin group McL.

The complete tables of marks of these groups have been determined by the methods described above. These tables form a library which is part of GAP.

In [Buekenhout and Rees 1988] the poset structure of the Mathieu group tex2html_wrap_inline4753 is determined. The poset structure of the sporadic simple Janko group tex2html_wrap_inline4755 is determined in [Pahlings 1987]. The poset structure of the sporadic simple Janko group tex2html_wrap_inline4757 is determined in [Pfeiffer 1991]. Informations about parts of the subgroup lattice of the sporadic simple McLaughlin group McL can be found in [Diawara 1987].

A rather incomplete collection of facts about subgroups of the Mathieu groups is given in [Greenberg 1973]. In his review (MR 50#4731), P. Fong points out that ``this can not remain the last work on the subject.''

Götz Pfeiffer Wed Oct 30 09:52:08 GMT 1996