Let G be a finite group. The table of marks of G arises from a characterization of the permutation representations of G by certain numbers of fixed points. It provides a compact description of the subgroup lattice of G and enables explicit calculations in the Burnside ring of G. In this article we introduce a method for constructing the table of marks of G from tables of marks of proper subgroups of G. An implementation of this method is available in the GAP language. These computer programs are used to construct the table of marks of the sporadic simple Mathieu group . The final section describes how to derive information about the structure of G from its table of marks via the investigation of certain Möbius functions and the idempotents of the Burnside ring of G. The appendix contains tables with detailed information about and other groups.
The concept of a table of marks of a finite group G was introduced by William Burnside in the second edition of his classic book Theory of Groups of Finite Order [Burnside 1911, chapter XII,]. This table provides a means to characterize the permutation representations of G up to equivalence. At the same time the table of marks describes in some detail the poset of all conjugacy classes of subgroups of G. It thereby provides a very compact description of the subgroup lattice of G.
Traditionally, the computation of the table of marks of G starts by constructing the complete subgroup lattice of G. The table of marks of G then is derived mainly by counting inclusions between different conjugacy classes of subgroups of G. This method [Felsch and Sandlöbes 1984] is implemented in several computer systems and works for groups up to a certain size.
The purpose of this article is to introduce a method for the construction of the table of marks which is independent of the knowledge of the complete subgroup lattice of G and therefore can be applied to groups G which are too big to compute their complete subgroup lattice. The main idea of this approach is to use the ``known'' tables of marks of subgroups of G and to induce them to G in order to determine the table of marks of G. So the input of this method basically consists of the tables of marks of the maximal subgroups of G. In order to combine these tables into the complete table of marks of G we have to determine the fusion maps from the sets of conjugacy classes of subgroups of the maximal subgroups of G to the set of conjugacy classes of subgroups of G. Such a map associates to the M-conjugacy class of a subgroup U of a maximal subgroup M of G its G-conjugacy class.
These fusion maps can to a large extent be recovered from the tables of marks of the maximal subgroups of G by elementary group theory. We will approximate the fusion maps step by step using this information and, now and then, taking advantage of selected bits of additional information about G until the approximation process eventually stops with the correct fusion maps.
Additional information about the group G stems for example from the character table of G. If G has a small permutation representation we can use explicit representatives of the conjugacy classes of subgroups of the maximal subgroups of G to derive additional information. It is, however, not necessary to have explicit embeddings of the maximal subgroups into G. But, any source of information is welcome if it can answer the questions that arise in the approximation process. The point is that the number of open questions is hopefully small in comparison to the size of the group.
Computer programs which perform this method interactively have been implemented in the GAP language [Schönert et al. 1992]. This package of functions is available from the author.
With this method we can in particular determine the number of subgroups of the simple Mathieu group of order .
The construction of the table of marks of from which the above number of subgroups is derived is described in some detail in section 6. This group provides an extensive example of how the computer programs can be used. The list of all conjugacy classes of subgroups of is not printed here. The complete list of the classes together with their maximal subgroups, their minimal overgroups and their normalizers has 82 pages [Pfeiffer 1995] and is available from the author.
Section 1 recalls basic properties of finite group actions, introduces the table of marks and describes its relation to the Burnside ring of a finite group.
In section 2 the concept of induction of marks is described and the basic induction formula is proved in Theorem 2.2.
The next sections describe the method used to determine the fusion maps from the maximal subgroups into the given group. First, in section 3, we use the projective special linear group as a rather informal example to illustrate the process and the problems it raises. This is formalized in section 4, which also provides the theoretical tools that govern the approximation process. Section 5 summarizes the method into a strategy.
In section 8 the table of marks of G is used to investigate the Möbius function of the subgroup lattice of G and the idempotents of the Burnside ring of G, together with their implications on the structure of G.
The appendix contains tables with more detailed information about the subgroups of and subgroups of other simple (and almost simple) groups.