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Möbius functions.

For any finite poset X, a Möbius function tex2html_wrap_inline4803 can be defined as inverse of the incidence relation tex2html_wrap_inline4805 via

displaymath4767

for all tex2html_wrap_inline4807 , where tex2html_wrap_inline4809 denotes the Kronecker delta. In particular, we have tex2html_wrap_inline4811 for all tex2html_wrap_inline4813 . 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 tex2html_wrap_inline4817 of all subgroups of G with inclusion as incidence. Denote its Möbius function by tex2html_wrap_inline4821 . The investigation of this case goes back to Hall's article [Hall 1936] where it has been intensely studied. For every tex2html_wrap_inline4823 the value tex2html_wrap_inline4825 can be derived from the table of marks of G (cf. [Pahlings 1993], Proposition 1).

  La902

In other words, tex2html_wrap_inline4837 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.

proof912

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 tex2html_wrap_inline4867 . Then

displaymath4771

where tex2html_wrap_inline4869 is the number of m-tuples of elements of G which generate G and we have (cf. [Hall 1936], (3.3))

  La924

proof931

In the case of tex2html_wrap_inline4899 we get the following values (cf. [Hall 1936] (4.4)).

tabular941

From these we get

displaymath4775

essentially different ways to generate tex2html_wrap_inline4961 by 2 elements.

Even more detailed questions can be answered from the table of marks of G. We can, for example, fix one element tex2html_wrap_inline4967 and ask for the number of elements tex2html_wrap_inline4969 with tex2html_wrap_inline4971 . For any tex2html_wrap_inline4973 let

displaymath4776

We have

displaymath4777

and

displaymath4778

By Möbius inversion we get

displaymath4779

where tex2html_wrap_inline4975 denotes the Möbius function of the subgroup lattice of G. In particular,

displaymath4780

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

displaymath4781

We finally get

displaymath4782

Hence the numbers tex2html_wrap_inline4985 can be computed from the table of marks for every tex2html_wrap_inline4987 (or even for any subgroup tex2html_wrap_inline4989 instead of tex2html_wrap_inline4991 ).

For the alternating group tex2html_wrap_inline4993 one obtains thus the following values tex2html_wrap_inline4995 .

tabular975

(This result again yields tex2html_wrap_inline5027 .)

The second relevant poset in this context is the poset tex2html_wrap_inline5029 of conjugacy classes of subgroups (where the class of a subgroup U is incident to the class of V if there is a tex2html_wrap_inline5035 such that tex2html_wrap_inline5037 ).

Denote by tex2html_wrap_inline5039 the Möbius function of tex2html_wrap_inline5041 .

Again, the values tex2html_wrap_inline5043 for tex2html_wrap_inline5045 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 tex2html_wrap_inline5051 .

In the case of tex2html_wrap_inline5053 we get the following values.

tabular980

The values of the functions tex2html_wrap_inline5135 and tex2html_wrap_inline5137 are related in the following way.

Thm986

This is obviously true for abelian groups G, since in that case the two posets tex2html_wrap_inline5145 and tex2html_wrap_inline5147 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 tex2html_wrap_inline5151 ). In  [Pahlings 1993] it is shown that the theorem holds for the projective special linear group tex2html_wrap_inline5153 , p a prime. There are, however, counterexamples:

In [Bianchi et al. 1990] it has been observed that, for tex2html_wrap_inline5157 , tex2html_wrap_inline5159 , while tex2html_wrap_inline5161 . Thus tex2html_wrap_inline5163 provides a counterexample for U = 1. In its general form, the formula of theorem does not hold for the simple groups tex2html_wrap_inline5167 , tex2html_wrap_inline5169 , and tex2html_wrap_inline5171 [Pahlings 1993].

Using the tables of marks of the groups in Table 4 one finds that the theorem also fails for the simple groups tex2html_wrap_inline5173 , tex2html_wrap_inline5175 , tex2html_wrap_inline5177 , tex2html_wrap_inline5179 and McL, in the latter also for U=1.


next up previous
Next: Idempotents in the Burnside Up: Applications. Previous: Applications.

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