Finite Soluble Groups Satisfying the Swap Conjecture

J Algebr Comb (2015) 42:907–915 DOI 10.1007/s10801-015-0610-5

Finite soluble groups satisfying the swap conjecture

Andrea Lucchini1

Received: 8 February 2015 / Accepted: 26 May 2015 / Published online: 9 June 2015 © Springer Science+Business Media New York 2015

Abstract For a d-generated ﬁnite group G, we consider the graph d (G) (swap graph) in which the vertices are the ordered generating d-tuples and in which two vertices (x1,...,xd ) and (y1,...,yd ) are adjacent if and only if they differ only by one entry. It was conjectured by Tennant and Turner that d (G) is a connected graph. We prove that this conjecture is true if G is a soluble group satisfying some extra conditions, for example if the derived subgroup of G has odd order or is nilpotent.

Keywords Generating graph · Swap conjecture · Soluble groups

Mathematics Subject Classiﬁcation 20D10 · 20F05 · 05C25

1 Introduction

Let G be a ﬁnite group, and let d(G) be the minimal number of generators of G. For any d integer d ≥ d(G), let Vd (G) ={(g1,...,gd ) ∈ G |g1,...,gd =G} be the set of all generating d-tuples of G. The generating d-tuples γ1 and γ2 aresaidtobeNielsen equivalent if there is a sequence of elementary Nielsen transformations leading from γ1 and γ2. By an elementary Nielsen transformation, here we mean a replacement in γ = ( ,..., ) ∈ ( ) , ( = ) −1 g1 gd Vd G of a component gi by gi g j g j gi i j or by gi .Inthe free group Fn of rank n ≤ d, every γ = (g1,...,gd ) ∈ Vd (Fn) is Nielsen equivalent to ( f1,..., fn, 1,...,1), where ( f1,..., fn) is a ﬁxed free basis for Fn [7]. Thus, the graph whose vertices are the elements of Vd (Fn) and in which two vertices are adjacent if and only if they are Nielsen equivalent is connected. In [9], Tennant and

B Andrea Lucchini [email protected]

1 Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy 123 908 J Algebr Comb (2015) 42:907–915

Turner introduced the notion of “swap equivalence”, which is wider than the notion of Nielsen equivalence. The d-uples γ1 and γ2 ∈ Vd (G) are said to be swap equivalent if there is a sequence of elementary swaps leading from γ1 to γ2. An elementary swap is thought of as a transformation changing one element of the sequence to an arbitrary element of G. The property of this equivalence relation can be encoded in the “swap graph” d (G): two vertices (x1,...,xd ), (y1,...,yd ) ∈ Vd (G) are adjacent in the swap graph if and only they differ only by one entry. Tennant and Turner proposed the conjecture that d (G) is connected (“swap conjecture”). In [8], it is proved that the free metabelian group of rank 3 does not satisfy this conjecture, but no counterexample is known in the class of finite groups. In [3], it was proved that the conjecture is true if d ≥ d(G) + 1. The case when d = d(G) is much more difficult. In [4], it is proved that the 2-generated finite soluble groups satisfy the swap conjecture and the methods used in the proof of this result are adapted in [3] to prove a weaker version of the swap conjecture in the soluble case, considering the graph d (G) whose vertices are the generating d-tuples and in which two vertices (x1,...,xd ), (y1,...,yd ) are adjacent if and only if there exists i ∈{1,...,d} such that xi = yi . In this paper, we prove that the swap conjecture is true if G is a finite group whose derived subgroup has odd order (so in particular it is true if G itself has odd order). Theorem 1 Let G be a finite group whose derived subgroup has odd order. If d ≥ d(G), then the swap graph d (G) is connected. Indeed, this is a corollary of a more general result. Theorem 2 Suppose that a finite soluble group G has the following property: if A is a nontrivial irreducible G-module G-isomorphic to a complemented chief factor of G, then | EndG (A)| > 2.Ifd≥ d(G), then the swap graph d (G) is connected. Another consequence of Theorem 2 is the following: Corollary 3 Let G be a finite group whose derived subgroup is nilpotent. If d ≥ d(G), then the swap graph d (G) is connected. In particular, by the previous corollary, the swap conjecture is true if G is a finite metabelian or supersoluble group. The proof of Theorem 2 depends on the solution of a combinatorial problem in linear algebra. Denote by Mr×s(F) the set of the r × s matrices with coefficients over the field F.Leta be a nonnegative integer, b a positive integer and let n = a + b. Let A ∈ Ma×n(F) with rank A = a. Moreover, let A be the set of matrices B ∈ Mb×n(F) with the property that A det = 0. B

We define a graph A whose vertices are the matrices in A and in which two vertices B1 and B2 are adjacent if and only if they differ only by one column. Theorem 4 If F is a finite field and |F| > 2, then the graph A is connected. Unfortunately, the previous theorem is not true if |F|=2. Consider, for example, the matrix A = 11 with coefficients over the field with two elements. In this case, the graph A consists only of two isolated vertices: B1 = 10 and B2 = 01. 123 J Algebr Comb (2015) 42:907–915 909

2 Proof of Theorem 4

In the particular case, when a = 0, our theorem is equivalent to the following lemma (and no assumption on |F| is needed): Lemma 5 Let V be a vector space of dimension n over the ﬁeld F and consider the graph whose vertices are the ordered sequences (u1,...,un) of n linearly independent vectors of V and in which two sequences (v1,...,vn) and (w1,...,wn) are adjacent if and only if they differ only by one entry. Then, is a connected graph.

Proof First, we prove the following claim:

Claim 1 If (u1,...,un) is a vertex of and σ ∈ Sym(n), then there is a path in joining (u1,...,un) and (u1σ ,...,unσ ). In order to prove this claim, it sufﬁces to show that if 1 ≤ i < j ≤ n, then there is a path in between (u1,...,ui ,...,u j ,...,un) and (u1,...,u j ,...,ui ,...,un). However, these vertices are joined by the following path

(u1,...,ui ,...,u j ,...,un), (u1,...,ui + u j ,...,u j ,...,un), (u1,...,ui + u j ,...,ui ,...,un), (u1,...,u j ,...,ui ,...,un), proving Claim 1. Next, we prove the following claim.

Claim 2 Let {v1,...,vn} and {w1,...,wn} be two bases of the vector space V and suppose that {v1,...,vi ,w1,...,wn−i } is a basis of V . Then, there exists j ∈{1,...,i} such that {v1,...,vj−1,vj+1,...,vi ,w1,...,wn−i+1} is a basis of V . In order to prove this claim, suppose that {v1,...,vi ,w1,...,wn−i } is a basis of V . There exists α1,...,αi ,β1,...,βn−i ∈ F with wn−i+1 = αr vr + βsws. 1≤r≤i 1≤s≤n−i

There exists j such that α j = 0 (otherwise wn−i+1 would be a linear combination of w1,...,wn−i ) hence v j belongs to the subspace of V spanned by the vectors v1,...,vj−1,vj+1,...,vi ,w1,...,wn−i+1: but then v1,...,vj−1,vj+1,...,vi ,w1,...,wn−i+1 is a generating set of V, and consequently a basis. Now, we can conclude the proof. By Claim 1, it sufﬁces to prove the connectivity of the graph whose vertices are the bases of V and in which two bases are adjacent if and only if their intersection has cardinality n − 1. This follows from iterated applications of Claim 2. ˜ Proof of Theorem 4 Clearly, A and A˜ coincide if A and A have the same row space, so we may substitute A with its reduced row echelon form. Moreover, the graph A 123 910 J Algebr Comb (2015) 42:907–915

˜ and the graph A˜ are isomorphic if A is obtained from A by applying a columns permutation. So it not restrictive to assume ⎛ ⎞ 10··· 0 a1,a+1 ··· a1,n ⎜ ⎟ ⎜01··· 0 a2,a+1 ··· a2,n ⎟ A := ⎜. . . . ⎟ . ⎝. . ··· . . ··· ⎠ 00··· 1 aa,a+1 ··· aa,n Let B = B1 ,..., Bn ∈ A (where with Bi we denote the i-th column of B). In the following, we will denote by Ir the identity r × r matrix and by 0r×s the zero matrix in the ring Mr×s(F). First, we prove the following claim: Claim 1 If det Ba+1 ··· Bn = 0, then there is a path in the graph A joining B to the matrix 0b×a Ib . The case a = 0 follows from Lemma 5.Sowemayassumea > 0. Let A∗ be the (a − 1) × (n − 1) matrix obtained by removing the first row and the first column from ∗ A and letB be the b × (n − 1) matrix obtained by removing the first column from A∗ B. If det = 0, then by induction the graph ∗ contains a path B∗ A ∗ C0 = B ,...,Cu = 0b×a−1 Ib .

Since ⎛ ⎞ 1 a , ··· a , 1 2 1 n A det ⎝0 A∗ ⎠ = det = 0 0 Ci 0 Ci for each i ∈{0,...,u}, we can conclude that ∗ B, 0 B , 0 C1 , ··· , 0 Cu = 0b×a Ib A∗ is a path in the graph A. Now assume det ∗ = 0. In this case, in order to conclude B ˜ ˜ ˜ the proof of the claim, it sufﬁces to prove that there exists a matrix B = B1,...,Bn which is adjacent to B in A and has the properties ˜ = ∈{ + ,..., }; (1) B j B jfor each j a 1 n A∗ (2) det = 0. B˜ ∗ Let ∗ ∗ 0 A Ai U1 = , ∗ = U2 ···Un with Ui = . B1 B Bi 123 J Algebr Comb (2015) 42:907–915 911 A A∗ 0 A∗ Since det = 0 and det ∗ = 0, we have rank ∗ = n − 1 and B B B1 B A∗ rank ∗ = n−2. In particular, U1 ∈/ U2,...,Un and there exists (μ2,...,μn) = B ( ,..., ) ∈ n−1 μ = . μ = 0 0 F such that 2≤i≤n i Ui 0 It cannot be j 0 for each j ∈{2,...,a}; otherwise, the vectors Ua+1,...,Un would be linearly dependent, in contradiction with det Ba+1 ··· Bn = 0. So there exists i ∈{2,...,a} such that Ui = λ j U j with λ j ∈ F. j∈{/ 1,i}

Since |F| > 2, there exists α ∈ F \{0}, such that β =−α − λ j α1, j = 0. j∈{/ 1,i}

˜ Now, consider Ui = Ui + αU1. We have ⎛ ⎞ 1 ··· 0 ··· 0 a1,a+1 ··· a1,n ⎝ ··· ∗ ··· ∗ ∗ ··· ∗ ⎠ det 0 Ai Aa Aa+1 An B ··· B + αB ··· B B + ··· B 1⎛ i 1 a a 1 ⎞n 1 ··· β ··· 0 a1,a+1 ··· a1,n = ⎝ ··· ··· ∗ ∗ ··· ∗ ⎠ = det 0 0 Aa Aa+1 An 0 B1 ··· 0 ··· Ba Ba+1 ··· Bn ˜ so B and B = B1 ··· Bi + αB1 ··· Ba Ba+1 ··· Bn are adjacent vertices of A. ˜ On the other hand, Ui is not a linear combination of U2,...,Un and so ∗ ∗ ··· ∗ ··· ∗ A A2 Ai An det ˜ ∗ = det = 0. B B2 ··· Bi + αB1 ··· Bn

The next step in the proof will be to prove the following claim: ··· = < ˜ Claim 2 If rank Ba+1 Bn c b, then there exists a matrix B which is adjacent ˜ ˜ to B in A and has the property that rank Ba+1 ···Bn > c. Assume, for example, that Bn is a linear combination of Ba+1,...,Bn−1. We want to ∈ b prove that there exists Y F such that ··· A1 An−1 An = , (1) det ··· 0 B1 Bn−1 Y (2) rank Ba+1 ··· Bn−1 Y > c. ∼ b Let Y ∈ Mb×1(F) = F . If Y does not satisfy (1), then there exist λ1,...,λn−1 ∈ F such that An A1 An−1 = λ1 +···+λn−1 . Y B1 Bn−1 123 912 J Algebr Comb (2015) 42:907–915 = λ (λ ,...,λ ) Hence, Y 1≤i≤n−1 i Bi where 1 n−1 is a solution of the linear system ⎛ ⎞ λ 1 ⎜ . ⎟ A1 ··· An−1 ⎝ . ⎠ = An. λn−1 Let q =|F|. Since rank A1 ··· An−1 = a, the number of solutions of this linear system is at most qn−1−a = qb−1. But then there are at most qb−1 vectors Y ∈ Fb which do not satisfy (1). If Y does not satisfy (2), then Y ∈ W, being W the vector subspace b of F spanned by Ba+1,...,Bn−1. Since dim W = rank Ba+1 ··· Bn−1 = c < b, |W|=qc ≤ qb−1, and therefore, there are at most qb−1 vectors Y ∈ Fb which do not satisfy (2). But then there are at least qb − 2qb−1 vectors Y in Fb satisfying (1) and (2). Since q > 2, we have qb − 2qb−1 > 0 and the proof of Claim 2 is completed. Iterated applications of Claim 2 lead to the conclusion that either rank Ba+1 ··· Bn = b ˜ ˜ ··· ˜ = . or B is connected to a matrix B with rank Ba+1 Bn b In both cases, by Claim 1, B is connected to 0b×a Ib .

3 Irreducible soluble linear groups

We begin the section by recalling the following theorem due to Gaschütz [6].

Proposition 6 Let π be an epimorphism from a ﬁnite group T1 to T2. Assume that T1 can be generated by d elements and let z1,...,zd be d elements which generate T2; then, there exist y1,...,yd in T1 that generate T1 and π(yi ) = zi for every i = 1,...,d.

Now, let V be a finite dimensional vector space over a finite field of prime order. Let H be a d-generated linear soluble group acting irreducibly and faithfully on V and fix a generating d-tuple (h1,...,hd ) of H. For a positive integer u, we consider u the semidirect product Gu = V H where H acts in the same way on each of the u direct factors. Put F = EndH (V ). Let a be the dimension of V over F. We may identify H =h1,...,hd with a subgroup of the general linear group GL(a, F). In this identification, hi becomes an a × a matrix Xi with coefficients in F; denote by Ai the matrix Ia − Xi . Let u wi = (vi,1,...,vi,u) ∈ V . Then, every vi, j can be viewed as a 1 × a matrix. Denote the u × a matrix with rows vi,1,...,vi,u by Bi . The following result is proved in [2, Section 4].

Proposition 7 The group G = V u × H can be generated by d elements if and only if u ≤ a(d − 1). Moreover, 123 J Algebr Comb (2015) 42:907–915 913 ··· = . (1) rank A1 Ad a ··· u A1 Ad (2) h1w1,...,hd wd =V H if and only if rank = a + u. B1 ··· Bd d Now, ﬁx (h1,...,hd ) ∈ H such that H =h1,...,hd and let u ≤ a(d − 1). We deﬁne a graph (u, h1,...,hd ) in which the vertices are the ordered d-tuples u d u (w1,...,wd ) in (V ) with h1w1,...,hd wd =V H and in which two vertices (x1,...,xd ) and (y1,...,yd ) are adjacent if and only if they differ only by one entry.

Lemma 8 If |F|=q = 2 and u ≤ a(d − 1), then the graph (u, h1,...,hd ) is connected.

Proof By Proposition 6, the group epimorphism π : Gn(d−1) → Gu induces a graph epimorphism from (a(d −1), h1,...,hd ) to (u, h1,...,hd ). Therefore, it sufﬁces to prove our lemma in the case (a(d − 1), h1,...,hd ). Let A = A1 ··· Ad . By Proposition 7, the graph (a(d −1), h1,...,hd ) is isomorphic to the graph A whose A vertices are the block matrices B = B ··· B with the property that det = 1 1 d B and two of these block matrices are adjacent if and only if they differ only by one block. This graph A has the same vertices as the graph A described in the previous section, and clearly if B1 and B2 are adjacent in A, then they are also adjacent in A. But then the connectivity of A follows from Theorem 4.

4 Proof of Theorem 2

Given a subset X of a finite group G, we will denote by dX (G) the smallest cardinality of a set T of elements of G with the property that G =X, T . Lemma 9 [4, Lemma 2] Let X be a subset of G and N a normal subgroup of G and suppose that g1,...,gr , X, N=G. If r ≥ dX (G), we can find n1,...,nr ∈ Nso that g1n1,...,gr nr , X=G. Let now G be a finite soluble group. Given an irreducible G-group A which is G-isomorphic to a complemented chief factor of G,letRG (A) be the smallest normal subgroup contained in CG (A) with the property that CG (A)/RG (A) is G-isomorphic to a direct product of copies of A, and it has a complement in G/RG (A). The factor group CG (A)/RG (A) is called the A-crown of G. The nonnegative integer δG (A) ∼ δ (A) defined by CG (A)/RG (A) =G A G is called the A-rank of G, and it coincides with the number of complemented factors in any chief series of G that are G-isomorphic to A (see, for example, [1, Section 1.3]). Lemma 10 [1, Lemma 1.3.6] Let G be a finite soluble group with trivial Frattini subgroup. There exists a crown C/R and a nontrivial normal subgroup U of G such that C = R × U. Lemma 11 [5, Proposition 11] Assume that G is a finite soluble group with trivial Frattini subgroup and let C, R, U be as in the statement of Lemma 10.IfHU = HR = G, then H = G. 123 914 J Algebr Comb (2015) 42:907–915

Lemma 12 Let Frat(G) be the Frattini subgroup of G. If G/ Frat(G) satisﬁes the swap conjecture, then G also satisﬁes the swap conjecture

Proof It sufﬁces to notice that if (x1,...,xd ) ∈ Vd (G) and f1,..., fd ∈ Frat(G), then

(x1,...,xd ), (x1 f1, x2,...,xd ),...,(x1 f1, x2 f2,...,xd−1 fd−1, yd ), (x1 f1,...,xd fd )

is a path in d (G).

Proof of Theorem 1 We prove the theorem by induction on the order of G.IfG = 1, then d (G) contains only the vertex (1,...,1) and is connected. Now, assume G = 1. Choose two vertices (x1,...,xd ) and (y1,...,yd ) of d (G). First, we prove the following claim:

Claim A Let N be a nontrivial normal subgroup of G: then there exist n1,...,nd in N such that (x1,...,xd ) and (y1n1,...,yd nd ) are connected by a path in d (G). Let g = gN. By induction, there exist l ∈ N and gij ∈ G, 0 ≤ i ≤ l,1≤ j ≤ d such that

(x1,...,xd ) = (g01, ··· , g0d ), (g11,...,g1d ), ··· ,(gl1,...,gld) = (y1,...,yd ) is a path in the graph d (G/N). To conclude the proof of the claim, it sufﬁces to prove, by induction on j, that for each 1 ≤ j ≤ l there exist n j1,...,n jd ∈ N such that d (G) contains a path joining (x1,...,xd ) and (g j1n j1,...,g jdn jd). Assume that this statement has been proved for j. By assumption, there exists k ∈{1,...,d} such that g ji = g j+1,i for each i = k. Let X ={g jin ji | i = k}:since dX (G) ≤ d − 1 and G =X, g j+1,kN, by Lemma 9 there exists n j+1,k ∈ N such that

G =X, g j+1,kn j+1,k=g jin ji, g j+1,kn j+1,k | i = k.

If we set n j+1,i = n j,i for every i = k, we conclude that (g j1n j1,...,g jdn jd) and (g j+1,1n j+1,1,...,g j+1,d n j+1,d ) are adjacent in d (G), proving the claim. By Lemma 12, we may assume that Frat(G) = 1. In this case, by Lemma 10, there exists a crown C/R of G and a normal subgroup U of G such that C = R × U. We ∼ δ have R = RG (A) where A is an irreducible G-module and U =G A for δ = δG (A). By Claim A, there exist u1,...,ud in U such that d (G) contains a path joining (x1,...,xd ) and (y1u1,...,yd ud ). Therefore, to conclude our proof it sufﬁces to ﬁnd in d (G) a path between the two vertices (y1,...,yd ) and (y1u1,...,yd ud ). Consider the factor group G/R. We have C/R = UR/R =∼ U =∼ Aδ and either ∼ ∼ δ ∼ A = C p is a trivial G-module and G/R = (C p) or G/R = C/R H/R where H/R acts in the same say on each of the δ factors of C/R =∼ Aδ and this action is faithful and irreducible. Since G/R is d-generated, we have δ ≤ d if A is a trivial G-module, δ ≤ ( − ) d 1 dimEndG (A) A otherwise. But then we may apply the results obtained in Sect. 3 to the factor group G/R . More precisely, Lemma 5 if A is a trivial G-module, Lemma 8 otherwise, ensures that there exist uij ∈ U, 0 ≤ i ≤ l, 1 ≤ j ≤ d such that: 123 J Algebr Comb (2015) 42:907–915 915

(1) G =y1ui1,...,yd uidR for each 0 ≤ i ≤ l; (2) u01 = 1,...,u0d = 1; (3) ul1 = u1,...,uld = ud ; (4) for each i ∈{0,...,l − 1}, there exists ki ∈{1,...,d} with uij = ui+1, j for each j = ki .

For every 0 ≤ i ≤ l,wehavey1ui1,...,yd uidR = G and y1ui1,...,yd uidU = y1,...,yd U = G hence, by Lemma 11, y1ui1,...,yd uid=G. We conclude that

(y1u01,...,yd u0d ), (y1u11,...,yd u1d ),...,(y1ul1,...,yd uld) is a path between the two vertices (y1,...,yd ) and (y1u1,...,yd ud ).

Proof of Theorem 2 Let G be the derived subgroup of G. If |G | is odd, then G is soluble by the Feit–Thompson theorem, and consequently G is also soluble. Moreover, if X and Y are normal subgroups of G such that A = X/Y is a nontrivial irreducible

G-module, then |A| is a power of a prime divisor p of |G | and F = EndG (A) is a ﬁnite ﬁeld of characteristic p. Hence, |F|≥p ≥ 3 and we may apply Theorem 2.

Proof of Corollary 3 By Lemma 12, we may assume Frat(G) = 1. This means that G = M H where H is abelian and M = V1 ×···×Vu is the direct product of u irreducible nontrivial H-modules V1,...,Vu. Let Fi = EndH (Vi ) = EndG (Vi ):for ∈{ ,..., }, = . each i 1 u Vi is an absolutely irreducible H-module so dimFi Vi 1 Now assume that A is a nontrivial irreducible G-module G-isomorphic to a complemented ∼ chief factor of G:itmustbeA =G Vi for some i,so| EndG (A)|=|Fi |=|Vi |=|A|. It cannot be |A|=2, otherwise A would be a trivial G-module. So we may apply Theorem 1.

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