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## Möbius functions.

For any finite poset X, a Möbius function can be defined as inverse of the incidence relation via

for all , where denotes the Kronecker delta. In particular, we have for all . This is a natural generalization of the well known Möbius function in number theory, where the partial order is given as divisibility (cf. [Rota 1964]).

Two posets are in a natural way associated to a given finite group G: one is the lattice of all subgroups of G with inclusion as incidence. Denote its Möbius function by . The investigation of this case goes back to Hall's article [Hall 1936] where it has been intensely studied. For every the value can be derived from the table of marks of G (cf. [Pahlings 1993], Proposition 1).

In other words, is the entry in the column corresponding to U of the final row of the inverse of the unweighted table of marks, i.e. the matrix which is derived from the table of marks by dividing each row by its diagonal value.

The knowledge of these values of the Möbius function is sufficient in many applications. They determine, for example, the number of essentially different ways in which G can be generated by m elements. Denote this number by . Then

where is the number of m-tuples of elements of G which generate G and we have (cf. [Hall 1936], (3.3))

In the case of we get the following values (cf. [Hall 1936] (4.4)).

From these we get

essentially different ways to generate by 2 elements.

Even more detailed questions can be answered from the table of marks of G. We can, for example, fix one element and ask for the number of elements with . For any let

We have

and

By Möbius inversion we get

where denotes the Möbius function of the subgroup lattice of G. In particular,

where the last summation is over all representatives U of conjugacy classes [U] of subgroups of G. Note that by Lemma 1.3 (iv)

We finally get

Hence the numbers can be computed from the table of marks for every (or even for any subgroup instead of ).

For the alternating group one obtains thus the following values .

(This result again yields .)

The second relevant poset in this context is the poset of conjugacy classes of subgroups (where the class of a subgroup U is incident to the class of V if there is a such that ).

Denote by the Möbius function of .

Again, the values for can be computed from the table of marks. The incidence matrix of the poset of conjugacy classes of G is obtained from the table of marks by replacing every nonzero entry by 1. The inverse of this matrix contains the values of the Möbius function of the poset .

In the case of we get the following values.

The values of the functions and are related in the following way.

This is obviously true for abelian groups G, since in that case the two posets and coincide. For U = 1 the theorem has been proved in [Hawkes et al. 1989]. The generalization stated above is proved in [Pahlings 1993]. The theorem also holds for many non-solvable groups, (see [Bianchi et al. 1990], [Pahlings 1993], and, in particular, the above table for ). In  [Pahlings 1993] it is shown that the theorem holds for the projective special linear group , p a prime. There are, however, counterexamples:

In [Bianchi et al. 1990] it has been observed that, for , , while . Thus provides a counterexample for U = 1. In its general form, the formula of theorem does not hold for the simple groups , , and [Pahlings 1993].

Using the tables of marks of the groups in Table 4 one finds that the theorem also fails for the simple groups , , , and McL, in the latter also for U=1.

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