M\"obius function of the subgroup lattice of a finite group and Euler Characteristic

The M\"obius function of the subgroup lattice of a finite group has been introduced by Hall and applied to investigate several questions. In this paper, we consider the M\"obius function defined on an order ideal related to the lattice of the subgroups of an irreducible subgroup $G$ of the general linear group $\mathrm{GL}(n,q)$ acting on the $n$-dimensional vector space $V=\mathbb{F}_q^n$, where $\mathbb{F}_q$ is the finite field with $q$ elements. We find a relation between this function and the Euler characteristic of two simplicial complexes $\Delta_1$ and $\Delta_2$, the former raising from the lattice of the subspaces of $V$, the latter from the subgroup lattice of $G$.


Introduction
In this paper, motivated by recent and less recent results about the Möbius function µ for the subgroup lattice L(G) of a finite group G, we give a result which relates the Möbius function for a subgroup G of GL(n, q) to two simplicial complexes: one defined from the lattice of the subspaces fixed by a reducible subgroup H ≤ G and the second from the lattice L(G) of the subgroups of G.
We will introduce in the Preliminaries all the definitions and details useful for reading the paper.
In his PhD thesis [8], Shareshian considers the problem of computing µ(1, G) for several finite classical groups G; the idea is to approximate µ(1, G) through a good function f G,n,p (u, 1) , such that: (1) µ(1, G(n, p u )) = f G,n,p (u, 1) + Here, G = G(n, p u ) denotes a family of finite classical groups with the same defining classical form, which act in a natural way on the vector space V of finite dimension n over the finite field F q of order q = p u .If C 1 , . . ., C 8 , C 9 are the Aschbacher classes of maximal subgroups of a finite classical group (see [4]), C 9 is the class of almost simple groups not belonging to the first 8 classes of "geometric type" and the function f G,n,p (u, 1) provides an estimate of µ(1, G) with respect to the contributions given by the subgroups of G which belong to the classes C i , for i ∈ {1, . . ., 8}.
Actually, Shareshian's approach focuses on the first class C 1 (G), that is the class of reducible subgroups of G.
In particular, the reducible subgroups of G contribute to f G,n,p (u, 1) through the computation of the Möbius function of which is obtained by adjoining the maximum G to the order ideal In this paper, we will consider irreducible subgroups G of the general linear group GL(n, q), that is groups of linear automorphisms of a vector space V of dimension n over the finite field F q with q elements, which fix no non-trivial subspace of V .In this hypothesis, we will take a reducible subgroup H of G (that is, H fixes some proper subspace of V ) and we will work on the analogue of µ (see Sections 2 and 3 for all precise definitions).
The subject of this paper is somehow motivated by the following conjecture: Indeed, although the problem is still open in its general setting, it was reduced by Lucchini in [5] to the study of similar growth conditions for finite almost-simple groups.
The following Theorem is the main result of the present paper: Theorem 4.5.Consider a vector space V of finite dimension over F q .Let G be an irreducible subgroup of GL(V ) and H ≤ G. Then (4) and χ(∆ i ) (for i = 1, 2) are defined in the following Section 3, also for irreducible subgroups H.This will allow us to avoid the restriction to only reducible subgroups H of G in the statement of the Theorem.
In the final Section, we will use Theorem 4.5 to deal with µ(H, G) in some particular case.In [1], Theorem 4.5 is used to attack Conjecture 1.1 for some class of subgroups H of linear and projective groups.
We thank Andrea Lucchini and Johannes Siemons for many useful discussions.

Preliminaries
In this paper all the groups and sets are finite.
For main results about posets and lattices, we refer to [9].Here, we just recall some basic fact, useful for reading the paper.Definition 2.1.Let P be a finite poset.The Möbius function associated with P is the map µ P : P × P → Z satisfying µ P (x, y) = 0 unless x ≤ y , and defined recursively for x ≤ y by (5) µ P (x, x) = 1 and Notation.If P is the subgroup lattice L(G) of G, we will write µ(H, K) Definition 2.2.An (abstract) simplicial complex ∆ on a vertex set T is a collection ∆ of subsets of T satisfying the two following conditions: An element F ∈ ∆ is called a face of ∆, and the dimension of F is defined to be |F | − 1.In particular, the empty set ∅ is a face of ∆ (provided ∆ = ∅) of dimension −1.
Definition 2.3.The Euler characteristic χ(∆) of the simplicial complex ∆ is so defined: where F i denotes the number of the i-faces, i.e., of the faces of dimension i, in ∆.
We recall here the definition of an order ideal: Definition 2.4.Let (P, ≤) be a poset.An order ideal of P is a subset In particular, if A is a subset of P, then the set P ≤A := {s ∈ P | s ≤ a for some a ∈ A} ⊆ P is the order ideal of P generated by A.
Observe that a simplicial complex on a set of vertices T is just an order ideal of the boolean algebra B T that contains all one element subsets of T .Definition 2.5.Let ∆ be a simplicial complex, and χ(∆) be the Euler characteristic of ∆.The reduced Euler characteristic χ(∆) of ∆ is defined by χ(∅) = 0, and χ(∆) = χ(∆) − 1 if ∆ = ∅.See also Equation (3.22) in [9].
It is possible to build a simplicial complex from a poset P in the following way: The order complex ∆(P) of P is defined as the simplicial complex whose vertices are the elements of P and whose k-dimensional faces are the chains a 0 ≺ a 1 ≺ • • • ≺ a k of length k of distinct elements a 0 , . . ., a k ∈ P.
Now, denote by P the finite poset obtained from P by adjoining a least element 0 and a greatest element 1.The Möbius function µ P ( 0, 1) is related to χ(∆(P)) by a well-known result by Hall in [3] about the computation of µ P ( 0, 1) by means of the chains of even and odd length between 0 and 1.

3.
The ideal I(G, H) and the complexes ∆ i in Theorem 4.5 In this section, and in the rest of the paper, G is an irreducible subgroup of the general linear group GL(n, q) over the vector space V = F n q of finite dimension n over the finite field F q with q elements.We consider the natural action of G on the set of subspaces of V .
We define the order ideal I(G, H) and the simplicial complexes ∆ i (for i = 1, 2) that we consider in Theorem 4.5.In Remark 2 below, we will explicitly observe that the two complexes ∆ i are not the order simplicial complex rising from I(G, H).
Given a subgroup G of GL(n, q) and a subgroup H of G, put Definition 3.1.The reducible subgroup ideal in L(G) ≥H is the order ideal of L(G) ≥H generated by C(G, H) .Namely, Remark 2.Here we just note that the poset I(G, H) has already a minimum if H is reducible, so that µ I(G,H) (H, G) is not, in general, the reduced Euler characteristic of the order complex ∆(I(G, H)) of I(G, H).
To define the simplicial complexes ∆ i of Theorem 4.5, we begin with fixing some more notation.We denote by S(V, H) the lattice of H-invariant subspaces of V and define Moreover, given an irreducible group G ≤ GL(V ), and H ≤ G, we will consider the following three sets: Since for irreducible H we have that I(G, H) = {H, G}, then Theorem 4.5 is trivially verified in this case.
Definition 3.3.The simplicial complexes ∆ i of Theorem 4.5 are so defined: • The set of vertices T 1 is given by the subspaces W ∈ S(V, H) * for which H = stab G (W ); • the set of faces of ∆ 1 is given by Ψ ′ (G, H).
• The set of vertices T 2 is given by the subgroups M ∈ C(G, H) such that H = M ; • the set of faces of ∆ 2 is given by Ψ(G, H).
We explicitly observe what happens in the special case when, for some proper non-trivial subspace W of V , the subgroup H = stab G (W ) is maximal with respect to the property of being a stabilizer of a proper non-trivial subspace of V in G.In this case, we note that, by definition, T 1 = T 2 = ∅ and the set of faces of ∆ 1 and ∆ 2 is {∅}.Then Theorem 4.5 is trivially verified.

Computing µ I(G,H) (H, G)
In order to prove Theorem 4.5, we need the following Proposition 4.2 that gives a link between Ψ(G, H) and Ψ ′ (G, H), and also shows that the reduced Euler characteristics of the complexes ∆ i coincide.The proof of Proposition 4.2 follows at once from Lemma 4.1.
Lemma 4.1.Let T be a subgroup of a finite group L acting on a finite set X and let X ′ ⊆ X be a subset such that T ≤ L x for all x ∈ X ′ (as usual, L x denotes the stabilizer of x in L).Set is the disjoint union of all the S E .So, it suffices to show that for each E ∈ R the following identity is verified: For E = ∅, the identity ( 7) is trivially true.Now, fix a non-empty E ∈ R, and for each K ∈ E define and observe that Q can be represented as the following disjoint union: and we obtain the identity (7).Proposition 4.2.Let V be a vector space of finite dimension over F q and consider a subgroup H ≤ G ≤ GL(V ).We have:  Proposition 4.4.Let V be a vector space of finite dimension over F q .Let H ≤ G ≤ GL(V ).Then we have that

Conjecture 1 . 1 (
Mann, [6]).Let G be a PFG group and µ the Möbius function on the lattice of open subgroups of G. Then |µ(H, G)| is bounded by a polynomial function in the index |G : H| and the number of subgroups of G of index m with µ(H, G) = 0 grows at most polynomially in m.
H) and I(G, H) is empty .The subspaces fixed by H are said to be H-invariant.Definition 3.2.If H is reducible, we set I(G, H) := I(G, H) ∪ {G} by adjoining the maximum G to I(G, H), which has minimum H. Otherwise, if H is irreducible, we set I(G, H) := {H, G} by adjoining the minimum H and the maximum G to the empty poset ∅.
Proof.Consider the natural action of G on the set of subspaces of V .By Lemma 4.1 below, the equality follows at once from the definitions ofS(V, H) * , C(G, H), Ψ(G, H) and Ψ ′ (G, H) .With previous notation, we take T = H, L = G, R = Ψ(G, H), S = Ψ ′ (G, H).To prove Theorem 4.5, we need Proposition 4.4 which is achieved through Theorem 4.3 (Crosscut Theorem, see [9, Corollary 3.9.4])and Remark 4.