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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 7, Pages 2689–2703 S 0002-9947(01)02730-1 Article electronically published on March 12, 2001

ON THE SHELLABILITY OF THE ORDER COMPLEX OF THE SUBGROUP LATTICE OF A FINITE GROUP

JOHN SHARESHIAN
Abstract. We show that the order complex of the subgroup lattice of a finite group G is nonpure shellable if and only if G is solvable. A by-product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups.

1. Introduction

We will show that the order complex of the subgroup lattice of a finite group G is (nonpure) shellable if and only if G is solvable. The proof of nonshellability in the nonsolvable case involves the determination of the homotopy type of the order complexes of the subgroup lattices of many minimal simple groups.
We begin with some history and basic definitions. It is assumed that the reader is familiar with some of the rudiments of algebraic topology and finite group theory. No distinction will be made between an abstract simplicial complex ∆ and an arbitrary geometric realization of ∆. Maximal faces of a simplicial complex ∆ will be called facets of ∆.

Definition 1.1. A simplicial complex ∆ is shellable if the facets of ∆ can be

ordered σ1, . . . , σn so that for all 1 ≤ i < k ≤ n there exists some 1 ≤ j < k and x ∈ σk such that σi ∩ σk ⊆ σj ∩ σk = σk \ {x}. The list σ1, . . . , σn is called a shelling of ∆.

Note that this definition does not require that ∆ be pure. Pure shellable complexes are discussed in [1], and nonpure shellable complexes are introduced in [2].

Definition 1.2. Let P be a partially ordered set. The order complex ∆(P) is the simplicial complex whose k-simplices are chains of length k from P.

Note that a chain of length k from a poset P contains k + 1 elements. We will call P shellable if and only if ∆(P) is shellable.

Definition 1.3. Let G be a finite group.

(1) L(G) is the lattice of subgroups of G. (2) L(G) := L(G) \ {1, G}. (3) l(G) is the length of the longest chain in L(G).

Received by the editors February 18, 1999 and, in revised form, May 1, 1999.

1991 Mathematics Subject Classification. Primary 06A11; Secondary 20E15.

  • c
  • ꢀ2001 American Mathematical Society

2689

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  • JOHN SHARESHIAN

It follows immediately from Definition 1.1 that L(G) is shellable if and only if
L(G) is shellable. We will call G shellable if and only if L(G) is shellable. As stated above, we will prove the following theorem.

Theorem 1.4. A finite group G is shellable if and only if G is solvable.

The relation between subgroup lattices and pure shellability is already wellunderstood. Note that for any poset P, the order complex ∆(P) is pure if and only if P is graded, that is, every maximal chain in P has the same length. In [10], Iwasawa showed that for a finite group G, L(G) is graded if and only if G is supersolvable. In [14], Stanley introduced supersolvable lattices, which are lattices which share some important combinatorial properties with subgroup lattices of finite supersolvable groups. In [1], Bj¨orner showed that supersolvable lattices are shellable, thereby showing that the three conditions G is supersolvable, ∆(L(G)) is pure, and ∆(L(G)) is shellable and pure are equivalent.
It follows from Theorem 1.4 and the work of Bjo¨rner and Wachs in [2] that if
G is solvable then ∆(L(G)) has the homotopy type of a wedge of spheres. The following stronger result was obtained by Kratzer and Th´evenaz in [11].

Theorem 1.5 ([11, Corollaire 4.10]). Let G be a finite solvable group with chief

series 1 = G0 / G1 / . . . / Gr−1 / Gr = G. For 1 ≤ i < r, let mi be the number of complements to Gi/Gi−1 in G/Gi−1. Then ∆(L(G)) has the homotopy type of a

i=1

Q

wedge of m = r−1 mi spheres of dimension r − 2.

It follows that the only reduced homology group of ∆ = ∆(L(G)) which can be

e

nontrivial is Hr−2(∆), which is free of dimension m. In [16], Th´evenaz investigates the linear representation of G on this homology group which is determined by the action of G on ∆ by conjugation. In doing so, he obtains an even stronger result than Theorem 1.5. Namely, assume that m > 0 (where m is as in Theorem 1.5), and fix a chief series 1 = G0 / . . . / Gr = G. Let Γ(G) be the set of all chains 1 = Cr < Cr−1 < . . . < C0 = G such that each Ci is a complement to Gi in G. For γ ∈ Γ(G), let Sγ be the poset obtained by removing 1 and G from the sublattice of

S

L(G) generated by all the Gi and all the Ci. Let K(G) = γ∈Γ(G) Sγ. Th´evenaz shows (see Theorem 1.4 of [16]) that

• the identity embedding of K(G) into L(G) induces a homotopy equivalence of order complexes,

W

• ∆(K(G)) ' γ∈Γ(G) ∆(Sγ), and • for all γ ∈ Γ(G), ∆(Sγ) has the homotopy type of an (r − 2)-sphere.
So, the spheres in the given wedge are indexed by the chains γ of complements to the elements of a given chief series. Our proof that ∆(L(G)) is shellable when G is solvable uses the theory of recursive coatom orderings (see Section 5 of [2]), and we are unable to obtain the decomposition of Th´evenaz or even the numerical result of Kratzer and Th´evenaz as a corollary. However, Th´evenaz’ result suggests that the theory of EL-shellability (again, see section 5 of [2]) might be applied to the subgroup lattice of a solvable group, as in the following conjecture. As noted in [16], all chains γ ∈ Γ(G) are maximal chains in L(G).

Conjecture 1.6. Let G be a finite solvable group. Then L(G) admits an EL- labeling in which the falling maximal chains are exactly the elements of Γ(G).

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  • SHELLABILITY OF SUBGROUP LATTICES
  • 2691

I thank Anders Bj¨orner, Richard Lyons and Volkmar Welker for their encouragement and their helpful comments.

2. Homotopy equivalence of order complexes and properties of shellable complexes

It this section we will review some known results about order complexes and shellable complexes which will be used throughout the paper. The reader who is familiar with the main results in these areas can skip this section and refer back to it when necessary. We begin with some homotopy results of Quillen.

Definition 2.1. Let P, Q be posets.

(1) A morphism f : P → Q is a function such that x ≤P y implies f(x) ≤Q f(y). (2) If f : P → Q is a morphism, then for y ∈ Q define

fy := {x ∈ P : f(x) ≤ y}

and

fy := {x ∈ P : f(x) ≥ y} .

Proposition 2.2 ([12, 1.5]). If a poset P contains a unique minimal element or a

unique maximal element, then ∆(P) is contractible.

Proposition 2.3 ([12, Proposition 1.6]). Let f : P → Q be a morphism of posets.

If ∆(fy) is contractible for all y ∈ Q or ∆(fy) is contractible for all y ∈ Q, then f induces a homotopy equivalence between ∆(P) and ∆(Q).

Corollary 2.4. Let P be a finite poset. Set

S = {x ∈ P : x is covered by a unique element in P} and
T = {x ∈ P : x covers a unique element in P} .
Then for any S0 ⊆ S and T 0 ⊆ T , ∆(P) is homotopy equivalent to both ∆(P \ S0) and ∆(P \ T 0).

Proof. Let i : P \ S0 ,→ P be the identity embedding. Fix y ∈ P. If y ∈ S0 then iy = {x ∈ P \ S0 : x ≥ y}. If y ∈ S0, let z0 be the unique element of P covering y. For i ≥ 0, if zi ∈ S0 let zi+1 be the unique element of P covering zi. Since P is finite, there is a smallest k such that zk ∈ S0, and iy = {x ∈ P \ S0 : x ≥ zk}. In any case, ∆(iy) is contractible by Proposition 2.2, and i induces a homotopy equivalence of order complexes by Proposition 2.3. The proof for P \ T0 is similar.

Corollary 2.5. Let L be a finite lattice and let M be the sublattice of L consisting

  • ˆ
  • ˆ

of the minimum element 0, the maximum element 1, and all x ∈ L such that

ꢀ
V

ˆ ˆ

x = c∈C c for some set C of coatoms of L. For X ∈ {L, M} let X = X \ 0, 1 . Then ∆(L) and ∆(M) are homotopy equivalent.

Proof. Let i : M → L be the identity embedding, and let C be the set of coatoms

V

of L. For each x ∈ L, let C(x) = {c ∈ C : x ≤ c}. Then c∈C(x) c is the unique minimum element of ix. The corollary now follows from Propositions 2.2 and 2.3.

Next we state a well-known collapsibility result (see [8]).

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  • JOHN SHARESHIAN

Lemma 2.6. Let ∆ be a simplicial complex and let τ be a face of ∆ such that τ is properly contained in a unique facet of ∆. Let ∆τ be the complex obtained by removing from ∆ all faces containing τ. Then ∆ and ∆τ are homotopy equivalent.

The following results on shellable complexes appear in [2].
Definition 2.7. Let ∆ be a finite simplicial complex.
(1) For a simplex σ ∈ ∆, d(σ) := max {|τ| : σ ⊆ τ ∈ ∆}. (2) For 0 ≤ j ≤ i, fi,j is the number of simplices σ ∈ ∆ such that |σ| = j and d(σ) = i.

  • ꢁ
  • ꢂ

P

  • j
  • i−k

(3) For 0 ≤ j ≤ i, hi,j := k=0(−1)j−k

fi,k.

j−k

Definition 2.8. Let ∆ be a finite simplicial complex and let σ1, . . . , σn be a shelling of ∆.
(1) R(σk) := {x ∈ σk : σk \ {x} ⊆ σi for some i < k}. (2) Γj := {σk : |σk| = j and R(σk) = σk}.

Theorem 2.9 ([2, Theorems 3.4 and 4.1]). Let ∆ be a finite simplicial complex

and let σ1, . . . , σn be a shelling of ∆.
(1) Then hi,j is the number of σk such that |σk| = i and |R(σk)| = j. In particular, hj,j = |Γj| and hi,j ≥ 0 for all i, j.
(2) Also, ∆ has the homotopy type of a wedge of spheres, consisting of hj,j spheres

˜

of dimension j − 1 for each j. In particular, dim(Hj−1(∆)) = hj,j is at most the number of facets of ∆ having dimension j − 1.

Definition 2.10. Let ∆ be an abstract simplicial complex of finite dimension d. For each 0 ≤ r ≤ s ≤ d, define the complex

(r,s)

  • ∆
  • := {τ ∈ ∆ : dim(τ) ≤ s and τ ⊆ σ for some σ ∈ ∆ with dim(σ) ≥ r} .

Theorem 2.11 ([2, Theorem 2.9]). If ∆ is a shellable complex of finite dimension

d, then ∆(r,s) is shellable for all 0 ≤ r ≤ s ≤ d.

Definition 2.12. Let P be a finite poset with a unique minimum element 0 and a
ˆ
ˆunique maximum element 1. We say that P admits a recursive coatom ordering if

ꢀ

ˆ ˆ
P = 0, 1 or if there is an ordering m1, . . . , mr of the coatoms of P which satisfies

the following two properties.

  • ꢃ
  • ꢄ

ˆ
(R) For all j ∈ [r], the poset Pj = 0, mj admits a recursive coatom ordering in

  • ꢃ
  • ꢄ

ˆwhich the coatoms of Pj which are contained in a poset Pi = 0, mi for some

i < j come before all other coatoms.
(S) For all 1 ≤ i < k ≤ r and x ∈ P, if x ≤ mi and x ≤ mk, then there exist some j < k and some coatom w of Pk such that x ≤ w ≤ mj.

Since the posets we will work with are lattices, we can replace condition (S) in
Definition 2.12 by the equivalent condition described below. The equivalence of the given conditions follows immediately from the definition of a lattice.

Proposition 2.13. Let L be a finite lattice, and let m1, . . . , mr be an ordering of the coatoms of L. This ordering satisfies condition (S) of Definition 2.12 if and only if it satisfies the following condition.
(T) For all 1 ≤ i < j ≤ r there exists some k < j such that mi ∧ mj ≤ mk ∧ mj

  • ꢃ
  • ꢄ

ˆ

and mk ∧ mj is a coatom of 0, mj .

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  • SHELLABILITY OF SUBGROUP LATTICES
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The next result, which follows immediately from Theorems 5.8 and 5.11 of [2], shows that in order to prove that G is shellable it suffices to show that L(G) admits a recursive coatom ordering.

Theorem 2.14 (Bj¨orner-Wachs). Let P be a finite poset with a unique minimum

  • ˆ
  • ˆ

element 0 and a unique maximum element 1. If P admits a recursive coatom ordering, then P is shellable.

3. Shellable groups are solvable

In this section it will be shown that if a finite group G is shellable then G must be solvable. This result will be achieved by reducing the problem to the examination of the minimal simple groups. Recall that a minimal simple group is a nonabelian finite simple group all of whose proper subgroups are solvable. The reduction is achieved by applying an elementary result of Bjo¨rner on shellable posets. Recall that a section of a finite group G is a quotient group of a subgroup of G. If G is a nonsolvable finite group, then some section of G is a minimal simple group.

Proposition 3.1. If a finite group G is shellable, then every section of G is shellable.

Proof. Proposition 4.2 of [1] says that if a graded poset P is shellable then every interval in P is shellable. It is straightforward to adapt the proof of this proposition to the nongraded case. Now if N E H ≤ G then the interval [N, H] in L(G) is isomorphic to L(H/N), and our proposition follows immediately.

The goal of this section is achieved by proving the following theorem.

Theorem 3.2. A minimal finite simple group is not shellable.

The minimal finite simple groups were determined by Thompson, well before the classification of finite simple groups was achieved. A list is given below.

Theorem 3.3 ([17, Corollary 1]). A finite simple group all of whose proper sub-

groups are solvable is isomorphic to one of the groups listed below.

(A) L2(p) with p ≥ 5 prime and p ≡ 1, 4 mod 5.

(B) L2(2p) with p prime. (C) L2(3p) with p ≥ 3 prime. (D) Sz(2p) with p ≥ 3 prime.

(E) SL3(3).

  • ∼
  • ∼

  • =
  • Note that L (4) L (5) A , where A is the alternating group on five symbols.
  • =

  • 2
  • 2
  • 5
  • 5

The structures of the subgroup lattices of the minimal simple groups are wellunderstood, and Theorem 3.2 will be proved by careful examination of each of the five classes of groups given above. We begin by determining all nontrivial subgroups of G which are intersections of maximal subgroups when G is one of the groups described in cases (A) through (D). In what follows, Zn denotes a cyclic group of

a

order n and Dn denotes a dihedral group of order n. For a prime p, Ep denotes an elementary abelian group of order pa. For any groups X and Y , X.Y denotes a split extension of a group isomorphic to X by a group of automorphisms of X

  • ∼
  • ∼

  • =
  • isomorphic to Y . Also, a.b will denote X.Y if X

denote the symmetric and alternating groups on n letters, respectively.

Z and Y

  • Z . S and An
  • =

  • a
  • b
  • n

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  • JOHN SHARESHIAN

For a prime power q, all subgroups of L2(q) are listed in [4]. Those subgroups of L2(q) which are intersections of maximal subgroups are determined in [7]. The results when L2(q) is a minimal simple group are as follows.

Lemma 3.4. Let p > 7 be a prime such that L = L2(p) is a minimal simple group. The maximal subgroups of L are those on the following list.

|L|

  • • One conjugacy class of
  • Dp−1.

p−1

|L|

• One class of

Dp+1.

p+1

  • • One class of p + 1 p.p−1
  • .

2

|L|

• Two classes of

S4, if p ≡ 1, 7 mod 8.

24

|L|

• One class of

A4, if p ≡ 3, 5 mod 8.

12

Furthermore, the nontrivial nonmaximal subgroups of L which are intersections of maximal subgroups are those on the following list.

|L|

• One or two classes containing a total of

D8, if p ≡ 1, 7 mod 8.

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  • Math 602 Assignment 1, Fall 2020 Part A

    Math 602 Assignment 1, Fall 2020 Part A

    Math 602 Assignment 1, Fall 2020 Part A. From Gallier{Shatz: Problems 1, 2, 6, 9. Part B. Definition Suppose a group G operates transitively on the left of a set X. (i) We say that the action (G; X) is imprimitive if there exists a non-trivial partition Q of X which is stable under G. Here \non-trivial" means that Q 6= fXg and Q 6= ffxg : x 2 Xg : Such a partition Q will be called a system of imprimitivity for (G; X). (ii) We say that the action (G; X) is primitive if it is not imprimitive. 1. (a) Suppose that (G; X; Q) is a system of imprimitivity for a transitive left action (G; X), Y 2 Q, y 2 Y . Let H = StabG(Y ), K = StabG(y). Prove the following statements. • K < H < G and H operates transitively on Y . • jXj = jY j · jQj, jQj = [G : H], jY j = [H : K]. (These statements hold even if X is infinite.) (b) Suppose that (G; X) is a transitive left action, y 2 X, K := StabG(y), and H is a subgroup of G such that K < H < G. Let Y := H · y ⊂ X, and let Q := fg · Y j g 2 Gg. (Our general notation scheme is that StabG(Y ) := fg 2 G j g · Y = Y g.) Show that (G; X; Q) is a system of imprimitivity. (c) Suppose that (G; X) is a transitive left action, x 2 X. Show that (G; X) is primitive if and only if Gx is a maximal proper subgroup of G.
  • The M\" Obius Number of the Socle of Any Group

    The M\" Obius Number of the Socle of Any Group

    Contents 1 Definitions and Statement of Main Result 2 2 Proof of Main Result 3 2.1 Complements ........................... 3 2.2 DirectProducts.......................... 5 2.3 The Homomorphic Images of a Product of Simple Groups . 6 2.4 The M¨obius Number of a Direct Power of an Abelian Simple Group ............................... 7 2.5 The M¨obius Number of a Direct Power of a Nonabelian Simple Group ............................... 8 2.6 ProofofTheorem1........................ 9 arXiv:1002.3503v1 [math.GR] 18 Feb 2010 1 The M¨obius Number of the Socle of any Group Kenneth M Monks Colorado State University February 18, 2010 1 Definitions and Statement of Main Result The incidence algebra of a poset P, written I (P ) , is the set of all real- valued functions on P × P that vanish for ordered pairs (x, y) with x 6≤ y. If P is finite, by appropriately labeling the rows and columns of a matrix with the elements of P , we can see the elements of I (P ) as upper-triangular matrices with zeroes in certain locations. One can prove I (P ) is a subalgebra of the matrix algebra (see for example [6]). Notice a function f ∈ I(P ) is invertible if and only if f (x, x) is nonzero for all x ∈ P , since then we have a corresponding matrix of full rank. A natural function to consider that satisfies this property is the incidence function ζP , the characteristic function of the relation ≤P . Clearly ζP is invertible by the above criterion, since x ≤ x for all x ∈ P . We define the M¨obius function µP to be the multiplicative inverse of ζP in I (P ) .
  • (Ipad) Subgroups Generated by a Subset of G Cyclic Groups Suppo

    (Ipad) Subgroups Generated by a Subset of G Cyclic Groups Suppo

    Read: Dummit and Foote Chapter 2 Lattice of subgroups of a group (iPad) Subgroups generated by a subset of G Cyclic groups Suppose is a subgroup of G. We will say that is the smallest subgroup of G containing S if S and if H is a subgroup of G containing S then S. Lemma 1. If and 0 both satisfy this denition then = 0. (Proof below that S = S .) h i hh ii Theorem 2. Let G be a group and S a subset. Then there is a unique smallest subgroup of G containing S. First proof (not constructive). Let S = H: h i H a subgroup of G S\H This is the unique smallest subgroup of G containing S. Indeed, these are all subgroups so their intersection is a subgroup: if x; y S then x; y H for all H in the intersection, so 1 2 h i 2 1 xy H and x¡ H and 1 H for all these subgroups, so xy; x¡ ; 1 S proving S is a gro2up. 2 2 2 h i h i Moreover, S is the smallest subgroup of G containing S because if H is any subgroup of G containinhg Si then H is one of the groups in the intersection and so S H. h i Second proof: Let S be dened as the set of all products (words) hh ii a a a 1 2 N 1 where either ai S or ai¡ S and if N = 0 the empty product is interpreted as 1G. I claim this is the smal2lest subgro2up of G containing S.
  • On the Structure of Overgroup Lattices in Exceptional Groups of Type G2(Q)

    On the Structure of Overgroup Lattices in Exceptional Groups of Type G2(Q)

    On the Structure of Overgroup Lattices in Exceptional Groups of Type G2(q) Jake Wellens Mentor: Michael Aschbacher December 4, 2017 Abstract: We study overgroup lattices in groups of type G2 over finite fields and reduce Shareshi- ans conjecture on D∆-lattices in the G2 case to a smaller question about the number of maximal elements that can be conjugate in such a lattice. Here we establish that, modulo this smaller ques- tion, no sufficiently large D∆-lattice is isomorphic to the lattice of overgroups OG(H) in G = G2(q) for any prime power q and any H ≤ G. This has application to the question of Palfy and Pud- lak as to whether each finite lattice is isomorphic to some overgroup lattice inside some finite group. 1 Introduction Recall that a lattice is a partially ordered set in which each element has a greatest lower bound and a least upper bound. In 1980, Palfy and Pudlak [PP] proved that the following two statements are equivalent: (1) Every finite lattice is isomorphic to an interval in the lattice of subgroups of some finite group. (2) Every finite lattice is isomorphic to the lattice of congruences of some finite algebra. This leads one to ask the following question: The Palfy-Pudlak Question: Is each finite lattice isomorphic to a lattice OG(H) of overgroups of some subgroup H in a finite group G? There is strong evidence to believe that the answer to Palfy-Pudlak Question is no. While it would suffice to exhibit a single counterexample, John Shareshian has conjectured that an entire class of lattices - which he calls D∆-lattices - are not isomorphic to overgroup lattices in any finite groups.
  • Computing the Maximal Subgroups of a Permutation Group I

    Computing the Maximal Subgroups of a Permutation Group I

    Computing the maximal subgroups of a permutation group I Bettina Eick and Alexander Hulpke Abstract. We introduce a new algorithm to compute up to conjugacy the maximal subgroups of a finite permutation group. Or method uses a “hybrid group” approach; that is, we first compute a large solvable normal subgroup of the given permutation group and then use this to split the computation in various parts. 1991 Mathematics Subject Classification: primary 20B40, 20-04, 20E28; secondary 20B15, 68Q40 1. Introduction Apart from being interesting themselves, the maximal subgroups of a group have many applications in computational group theory: They provide a set of proper sub- groups which can be used for inductive calculations; for example, to determine the character table of a group. Moreover, iterative application can be used to investigate parts of the subgroups lattice without the excessive resource requirements of com- puting the full lattice. Furthermore, algorithms to compute the Galois group of a polynomial proceed by descending from the symmetric group via a chain of iterated maximal subgroups, see [Sta73, Hul99b]. In this paper, we present a new approach towards the computation of the conju- gacy classes of maximal subgroups of a finite permutation group. For this purpose we use a “hybrid group” method. This type of approach to computations in permu- tation groups has been used recently for other purposes such as conjugacy classes [CS97, Hul], normal subgroups [Hul98, CS] or the automorphism group [Hol00]. For finite solvable groups there exists an algorithm to compute the maximal sub- groups using a special pc presentation, see [CLG, Eic97, EW].