ON the SHELLABILITY of the ORDER COMPLEX of the SUBGROUP LATTICE of a FINITE GROUP 1. Introduction We Will Show That the Order C

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. <ul style="display: flex;"><li style="flex:1">c</li><li style="flex:1">ꢀ2001 American Mathematical Society </li></ul>2689 <a href="/goto?url=https://www.ams.org/journal-terms-of-use" target="_blank">License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use </a><ul style="display: flex;"><li style="flex:1">2690 </li><li style="flex:1">JOHN SHARESHIAN </li></ul>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], Björner 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 Thévenaz 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 Qwedge of m = r−1 mi spheres of dimension r − 2. It follows that the only reduced homology group of ∆ = ∆(L(G)) which can be enontrivial is Hr−2(∆), which is free of dimension m. In [16], Thévenaz 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 SL(G) generated by all the Gi and all the Ci. Let K(G) = γ∈Γ(G) Sγ. Thévenaz 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 Thévenaz or even the numerical result of Kratzer and Thévenaz as a corollary. However, Thévenaz’ 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). <a href="/goto?url=https://www.ams.org/journal-terms-of-use" target="_blank">License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use </a><ul style="display: flex;"><li style="flex:1">SHELLABILITY OF SUBGROUP LATTICES </li><li style="flex:1">2691 </li></ul>I thank Anders Björner, 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 <ul style="display: flex;"><li style="flex:1">ˆ</li><li style="flex:1">ˆ</li></ul>of the minimum element 0, the maximum element 1, and all x ∈ L such that ꢀ Vˆ 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 Vof 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]). <a href="/goto?url=https://www.ams.org/journal-terms-of-use" target="_blank">License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use </a><ul style="display: flex;"><li style="flex:1">2692 </li><li style="flex:1">JOHN SHARESHIAN </li></ul>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. <ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul>P<ul style="display: flex;"><li style="flex:1">j</li><li style="flex:1">i−k </li></ul>(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 õf 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) <ul style="display: flex;"><li style="flex:1">∆</li><li style="flex:1">:= {τ ∈ ∆ : dim(τ) ≤ s and τ ⊆ σ for some σ ∈ ∆ with dim(σ) ≥ r} . </li></ul>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 ûnique 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. <ul style="display: flex;"><li style="flex:1">ꢃ</li><li style="flex:1">ꢄ</li></ul>ˆ (R) For all j ∈ [r], the poset Pj = 0, mj admits a recursive coatom ordering in <ul style="display: flex;"><li style="flex:1">ꢃ</li><li style="flex:1">ꢄ</li></ul>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 <ul style="display: flex;"><li style="flex:1">ꢃ</li><li style="flex:1">ꢄ</li></ul>ând mk ∧ mj is a coatom of 0, mj . <a href="/goto?url=https://www.ams.org/journal-terms-of-use" target="_blank">License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use </a><ul style="display: flex;"><li style="flex:1">SHELLABILITY OF SUBGROUP LATTICES </li><li style="flex:1">2693 </li></ul>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 (Björner-Wachs). Let P be a finite poset with a unique minimum <ul style="display: flex;"><li style="flex:1">ˆ</li><li style="flex:1">ˆ</li></ul>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 subgroups 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). <ul style="display: flex;"><li style="flex:1">∼</li><li style="flex:1">∼</li></ul><ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">Note that L (4) L (5) A , where A is the alternating group on five symbols. </li><li style="flex:1">=</li></ul><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">2</li><li style="flex:1">5</li><li style="flex:1">5</li></ul>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 aorder 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 <ul style="display: flex;"><li style="flex:1">∼</li><li style="flex:1">∼</li></ul><ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">isomorphic to Y . Also, a.b will denote X.Y if X </li></ul>denote the symmetric and alternating groups on n letters, respectively. Z and Y <ul style="display: flex;"><li style="flex:1">Z . S and An </li><li style="flex:1">=</li></ul><ul style="display: flex;"><li style="flex:1">a</li><li style="flex:1">b</li><li style="flex:1">n</li></ul><a href="/goto?url=https://www.ams.org/journal-terms-of-use" target="_blank">License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use </a><ul style="display: flex;"><li style="flex:1">2694 </li><li style="flex:1">JOHN SHARESHIAN </li></ul>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| <ul style="display: flex;"><li style="flex:1">• One conjugacy class of </li><li style="flex:1">Dp−1. </li></ul>p−1 |L| • One class of Dp+1. p+1 <ul style="display: flex;"><li style="flex:1">• One class of p + 1 p.p−1 </li><li style="flex:1">.</li></ul>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.

ON the SHELLABILITY of the ORDER COMPLEX of the SUBGROUP LATTICE of a FINITE GROUP 1. Introduction We Will Show That the Order C

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