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The subgroups of \(M_{24}\), or how to compute the table of marks of a finite group. (English) Zbl 0895.20017
The table of marks, introduced by Burnside in 1911, provides a way of describing the lattice of subgroups of a group. Each ‘mark’ is the number of fixed points of the action of one subgroup, \(U\) say, on the cosets of another, \(V\) say. Since this is invariant under conjugation of either \(U\) or \(V\), the number of rows (and columns) of the table is equal to the number of conjugacy classes of subgroups.
Even for groups of moderate size, this can be a very large table, and its computation provides something of a challenge. The present paper describes a semiautomatic procedure for determining the table by induction from the maximal subgroups. The crucial step in the induction is to determine when two subgroups of two (not necessarily distinct) maximal subgroups are actually conjugate in the whole group – this is achieved by a process of successive approximation, during which some extra specific information about the group may be needed.
The example of \(M_{24}\) is described in some detail, and a list of other results appended. Finally some applications are briefly described. For example, the number of generating subsets of size \(m\) can be determined by a Möbius inversion – the author illustrates this by proving the well-known fact that \(A_5\) has exactly 19 conjugacy classes of pairs of generators.

MSC:
20D30 Series and lattices of subgroups
20C40 Computational methods (representations of groups) (MSC2010)
20D08 Simple groups: sporadic groups
20F05 Generators, relations, and presentations of groups
20E28 Maximal subgroups
Software:
GAP
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