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Solvable Quotients of Subdirect Products of Perfect Groups Are Nilpotent

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Solvable Quotients of Subdirect Products of Perfect Groups Are Nilpotent SOLVABLE QUOTIENTS OF SUBDIRECT PRODUCTS OF PERFECT GROUPS ARE NILPOTENT KEITH KEARNES, PETER MAYR, AND NIK RUSKUCˇ Abstract. We prove the statement in the title and exhibit examples of quo- tients of arbitrary nilpotency class. This answers a question by D. F. Holt. 1. Introduction A subgroup S of a direct product G1 × · · · × Gn is said to be subdirect if its projection to each of the factors Gi is surjective. Subdirect products have recently been a focus of interest from combinatorial and algorithmic point of view; see, for example, [1]. They are also frequently used in computational finite group theory, notably to construct perfect groups; see [6]. In September 2017 D. F. Holt asked on the group pub forum whether a solvable quotient of a subdirect product of perfect groups is necessarily abelian. The question was re-posted on mathoverflow [5] where Holt gave some background: \This problem arose in a study of the complexity of certain algorithms for finite permutation and matrix groups. The group S in the applications is the kernel of the action of a transitive but imprimitive permutation (or matrix) group on a block system. So in that situation the Gi are all isomorphic, and Aut(S) induces a transitive action on the direct factors Gi (1 ≤ i ≤ n)." In the present note we start from the observation that every solvable quotient of a subdirect product of perfect groups is nilpotent, proved in Section 2, and then proceed to construct several examples demonstrating that the nilpotency class can be arbitrarily high. Specifically, in Section 5 we exhibit an infinite group G such that for every d there is a subdirect subgroup of G2d with a nilpotent quotient of degree d. In Section 6 we exhibit a perfect subgroup G of the special linear group 4 SL6(F4) such that G contains a subdirect subgroup with a nilpotent quotient of class 2. This example is generalized in Section 7 to a perfect subgroup G of the 2n−1 special linear group SL2n(F4) such that G contains a subdirect subgroup with Date: July 23, 2018. 2010 Mathematics Subject Classification. Primary: 20F14, Secondary: 20E22. Key words and phrases. Subdirect product, subgroup, perfect group, nilpotent. The first two authors were supported by the National Science Foundation under Grant No. DMS 1500254. 1 2 K. KEARNES, P. MAYR, AND N. RUSKUCˇ a nilpotent quotient of class n − 1. These examples are all based on a particular subdirect product construction, which is described in Section 3, and whose lower central series is computed in Section 4. Throughout the paper we will use the following notation. Let N denote the set of positive integers. For n 2 N, we write [n] to denote f1; : : : ; ng. For elements x; y of a group G we write xy := y−1xy and [x; y] := x−1y−1xy. The higher commutators in a group G will always be assumed to be left associated, i.e. [x1; : : : ; xk] := [x1; x2]; : : : ; xk ; [N1;:::;Nk] := [N1;N2];:::;Nk for xi 2 G and Ni E G. Furthermore, for k ≥ 1 we let γk(G) := [G; : : : ; G]; γ1(G) := G; | {z } k denote the kth term in the lower central series of G and, for N E G and k ≥ 0, write [G; kN] := [G; N; : : : ; N]; [G; 0N] := G: | {z } k The following two facts about commutators for a group G with a normal subgroup N will frequently be used (k; l 2 N; m 2 N [ f0g): [C1] [γk(N); γl(N)] ≤ γk+l(N); [C2] [G; mN]; γl(N) ≤ [G; m+lN]. The first can be found in [8, 5.1.11(i)]. The second follows by induction on l. The case l = 1 holds by the definition of [G; kN] and the fact that γ1(N) = N. For l > 1 we have h i h i [G; mN]; γl(N) ≤ [G; mN];N ; γl−1(N) · [G; mN]; γl−1(N) ;N by the Three Subgroup Lemma h i = [G; m+1N]; γl−1(N) · [G; mN]; γl−1(N) ;N ≤ [G; m+lN] · [G; m+l−1N];N by induction = [G; m+lN]: 2. Solvable quotients must be nilpotent That solvable quotients of subdirect products of perfect groups are necessarily nilpotent is an easy consequence of the following more general observation. Theorem 2.1. Let N E S ≤sd G1 × · · · × Gn such that N; S are both subdirect products of groups G1;:::;Gn. Then S=N is nilpotency of class at most n − 1. QUOTIENTS OF SUBDIRECT PRODUCTS OF PERFECT GROUPS 3 Proof. For i = 1; : : : ; n, let Ki E S be the kernel of the projection of S onto Gi. Tn Notice that i=1 Ki = 1. Moreover we claim that KiN = S for all i = 1; : : : ; n: (1) For the inclusion ≥, let s 2 S. Since the projection of N onto Gi is onto by −1 assumption, we have b 2 N such that bi = si. Then sb is in Ki which proves (1). It now follows that γn(S) = [K1N; : : : ; KnN] ≤ [K1;:::;Kn]N: Tn Since [K1;:::;Kn] ≤ i=1 Ki = 1, we obtain γn(S) ≤ N. Remark 2.2. The analogue of Theorem 2.1 holds, with exactly the same proof, in any congruence modular variety when the normal subgroup N is replaced by a congruence ν of the subdirect product S such that the join of ν with any projection kernel ρi (i ≤ n) yields the total congruence on S. In fact, then S/ν is supernilpotent of class at most n − 1. We refer the reader to [3] for the definition of commutators of congruences and to [7] for supernilpotence and higher commutators in this general setting. It then follows that the analogue of Theorem 2.1 holds for rings, K-algebras, Lie algebras, and loops. Corollary 2.3. Let S ≤sd G1 × · · · × Gn be a subdirect product of perfect groups G1;:::;Gn, and let N E S be a normal subgroup of S. If S=N is solvable, then S=N is nilpotent of class at most n − 1. Proof. For i = 1; : : : ; n, let Ni denote the projection of N on Gi. Since each Gi is perfect and Gi=Ni is solvable by assumption, we obtain Gi = Ni and the result follows from Theorem 2.1. 3. A subdirect construction By Corollary 2.3, every solvable quotient of the subdirect product of two perfect groups is abelian. In what follows, we are going to show that in fact there exist subdirect powers of a perfect group G with quotients of arbitrarily large nilpotency class. To this end we introduce a construction that is based on higher commutator relations from universal algebra [7]. Let G be a group, and let d 2 N. We will be working in the direct product of 2d copies of G with components indexed by the power set P := P([d]). The set P will be linearly ordered by the short-lex ordering; specifically A <sl B iff jAj < jBj or jAj = jBj and min(A n B) < min(B n A) : P Now, for any A ⊆ [d], u 2 G and K ⊆ G, define ∆A(u) 2 G by u if A ⊆ B ∆ (u)(B) := A 1 otherwise. 4 K. KEARNES, P. MAYR, AND N. RUSKUCˇ and ∆A(K) := ∆A(u): u 2 K : The following is an immediate consequence of the above definition: Lemma 3.1. If K; L E G and A; B ⊆ [d], then ∆A(K); ∆B(L) = ∆A[B [K; L] : Now, for a normal subgroup N E G and for k 2 [d + 1] we define the following subset of GP : Y Y Γk := Γk(G; N) := ∆A [G; jAjN] · ∆A γjAj(N) : (2) jAj<k jA|≥k Here the factors in the products are ordered by short-lex on P. Lemma 3.2. Let A; B ⊆ [d]. Then (i) [G; jAjN]; [G; jBjN] ≤ [G; jA[BjN], (ii) [G; jAjN]; γjBj(N) ≤ [G; jA[BjN] for B 6= ; and (iii) γjAj(N); γjBj(N) ≤ γjA[Bj(N) for A; B 6= ;. Proof. Immediate from [C1] and [C2]. P Lemma 3.3. Γk(G; N) is a subdirect subgroup of G for each k 2 [d + 1] and N E G. Proof. First notice that all factors in the definition of Γk are subgroups. That Γk projects onto each coordinate follows from the presence of ∆; [G; 0N] = ∆;(G), the diagonal of GP . To prove that Γk is a subgroup, consider two generic factors ∆A(K) and ∆B(L) in its definition for A <sl B. That is, K = [G; jAjN] if jAj < k and K = γjAj(N) otherwise, and likewise for L. By Lemma 3.1, we obtain ∆B(L) · ∆A(K) = ∆A(K) · ∆B(L) · ∆B(L); ∆A(K) = ∆A(K) · ∆B(L) · ∆A[B [K; L] : By Lemma 3.2, we have [K; L] ≤ [G; jA[BjN] if jA [ Bj < k; further [K; L] ≤ γjA[Bj(N) if jA [ Bj ≥ k. Hence ∆A[B [K; L] is contained in another factor of the product defining Γk. Since B ≤sl A [ B, the result follows. Remark 3.4. In reference to Holt's motivation, we note that the automorphism group of S := Γ1 acts transitively on the index set P. More precisely, for i 2 [d], the permutation on P that maps a subset A of [d] to its symmetric difference with fig induces an automorphism ri on S. This can be easily seen by the action of ri on the generators ∆A(u) of S: ri(∆A(u)) = ∆A(u) if i 62 A; else −1 ri(∆A(u)) = ∆Anfig(u) · ∆A(u) .
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