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.
