Groups generated by transvections: a digraph theoretical approach

Lino Di Martino, Andrea Previtali

Roberto Radina

Udine, 20 November 2003

Transvections

Let K = 𝔽_{q} be a finite field of characteristic p and V = K^{n}. We say t ∈ GL(V) is a transvection if there exist c ∈ V and ϕ ∈ V^{∗} = hom(V, K) such that t(v) − v = ϕ(v)c for any v ∈ V and ϕ(c) = 0.

We identify V^{∗} with^{n}K (row). Thus ϕ(v) = av, for a given a ∈^{n}K. Notice that det(t) = 1.

We say a = a(t) and c = c(t) are the axis and the centre of t. These are determined up to a scalar. To avoid ambiguity we identify any transvection t with (c(t), a(t)).

The Problem

Given a finite set U of transvections in SL_{n}(q) determine G = ⟨U⟩.

Historical background

Theorem 1. SL_{n}(q) is generated by its set of transvections. Theorem 2 (Piper 1966-68). Let G be a subgroup of SL_{n}(q) not fixing any 𝔽_{p}-subspace and possessing two transvections with the same axis and non collinear centers. Then either G = SL_{n}(q) or p = 2 and G = 3.Alt(6) ≤ SL_{3}(4).

Given a transvection t and λ ∈ K, we set t^{λ} = 1 + λ(t − 1) and call t^{K} = {t^{λ} : λ ∈ K} the root subgroup generated by t. Theorem 3 (McLaughlin 1967). Let G ≤ SL_{n}(q), q ≠ 2, be generated by root subgroups. If O_{p}(G) = 1, then the natural module is completely reducible and G is a direct product of special linear or symplectic groups over 𝔽_{q}.

Historical background II

Theorem 4 (McLaughlin 1969). Let G be a proper irreducible subgroup of SL_{n}(2) generated by transvections. Then n > 3 and G is Sp_{n}(2), O_{n}^{ε}(2), Sym(n + 2) or Sym(n + 1). Theorem 5 (Zalesskii, Serežkin 1976). Let G be an irreducible subgroup of SL_{n}(K), with p, n > 2, generated by root subgroups. Then G is conjugate in GL_{n}(K) to SL_{n}(F), Sp_{n}(F) or SU_{n}(F), for some F ≤ K.

We now drop the assumption that t ∈ G implies t^{K} ≤ G. Theorem 6 (Wagner 1974). Let G ≤ SL_{n}(K) be irreducible, generated by transvections, and there exists a transvection t ∈ G such that |t^{K} ∩ G| = p^{k} > 2. Then G is SL_{n}(p^{k}), Sp_{n}(p^{k}), or SU_{n}(p^{2k}).

The Humphries matrix

Given s, t transvections we define

s ∗ t := a(s)c(t).

Let U = (t_{1}, …, t_{h}) be a list of transvections in SL_{n}(K). Then we define

• A(U) ∈ (K)_{h×n} as the axis matrix;

• C(U) ∈ (K)_{n×h} as the centre matrix;

• H(U) = A(U)C(U) = (t_{i} ∗ t_{j}) ∈ (K)_{h×h} as the Humphries matrix.

Digraphs

To a set U of transvections in SL_{n}(K), we associate a labelled digraph Γ(U), with labels in K.

Its vertices are the elements of U and s is connected to t iff s ∗ t ≠ 0. If so s ∗ t will be the label of such arc.

We say that the digraph Γ is (strongly) connected if for any pair of vertices s, t ∈ Γ there exists a path from s to t.

We say that Γ is one-way if there exist some vertices s, t and the arc from s to t but no arc from t to s.

t-Equivalence Relation

Substitution of t with s^{−λ}ts^{λ}, for some s, t ∈ U, λ ∈ K is called an elementary t-transformation. Humphries says that V is t-equivalent to U, if V is obtained applying a finite number of elementary t-transformations to U. Theorem 7 (Humphries 1986). Let U be a set of n > 2 transvections in SL(n, K). Then U^{K} generates SL_{n}(K) iff

1. det(H(U)) ≠ 0;

2. Γ(U) is connected;

3. Γ(U) is t-equivalent to a one-way digraph.

Subfields

Given a digraph Γ(U) with U a transvection set in SL(n, K) and g an integer, we denote with

L_{g}(U)

the subfield of K generated by the product of labels along all closed paths of length at most g. We set \underline{L}(U) := L_{2}(U) and \overline{L}(U) := L_{∞}(U). Clearly L_{g}(U) ↑ L_{∞}(U).

Cyclic graphs

Lemma 1 (Dickson). Let G be the subgroup of SL_{2}(K) generated by T_{12}(1), T_{21}(δ). Set p := charK and F = 𝔽_{p}[δ]. Then

1. G = SL_{2}(F) or

2. δ^{2} = −1, p = 3, and G ≃ SL_{2}(5) or

3. p = 2 and G is dihedral.

If h > 2 no restriction on p is needed. We say the digraph Γ is a cycle if t_{i} is linked to t_{j} iff j = (i mod h) + 1, where |Γ| = h. Theorem 8 (Cycle). Let Γ(U) be a cycle of length h > 2 and label product δ, U ⊆ SL_{n}(K). Then ⟨U⟩ is GL_{n}(K)-conjugate to ≃ SL(h, F) ⊕ I_{n−h}, where F = 𝔽_{p}[δ].

Weak t-equivalence

We do not allow unnecessary scalars Definition 2. We say U ∼_{1} V if V = U ∪ {t^{s}}, for t, s ∈ U;

We say U ∼_{2} V if V = U ∪ {t^{λ}}, for t ∈ U and λ ∈ \overline{L}(U).

We call ≈ the symmetric and transitive closure of the union of ∼_{1} and ∼_{2}.

We show that ≈ does not bring us out of G. Proposition 3 (No unnecessary scalar). Let char(K) ≠ 2, 3, |U| > 1, Γ(U) connected and Γ(Ũ) one-way. If t ∈ U and λ ∈ \overline{L}(U) then t^{λ} ∈ G.

Normal Form

Theorem 9. Assume char(K) ≠ 2, 3, Γ(U) connected, Γ(Ũ) one-way ande set r := rank(H(V)). Then U ≈ V, where V = {s_{1}, …, s_{r}, t_{1}, …, t_{k}}, Γ(s_{1}, …, s_{r}) is a cycle, any t is connected to some s, s_{1} excluded, only s_{r} is connected to some t hence to all of them. The t’s are not linked. Corollary 1. With the above assumptions \underline{L}(Ũ) = \overline{L}(U).

Main Theorem

Theorem 10. Let char(K) = p ≠ 2, 3 and r = rank(H(U)). Suppose that Γ(U) is connected, Γ(Ũ) is a one-way digraph. Then:

1. \underline{L}(Ũ) = \overline{L}(U);

2. G = N ⋉ S, where N = O_{p}(G) and S ≃ SL(r, \overline{L}(U)), r > 2;

3. N = 1 iff A(U), C(U), and H(U) have the same rank.

Up to conjugation G has shape

( \begin{matrix}I & x & z \\ 0 & s & y \\ 0 & 0 & I \end{matrix} ) ,

where the dimensions of the blocks depend on rank(A(U)) and rank(C(U)) and s varies in SL_{r}(\overline{L}(U)). Thus N has p-lower nilpotency class at most 2.

Transvections in classical groups

Let (V, f) be a K-space with a non-degenerate form f and I(V) the corresponding isometries, then a transvection t = (ϕ, c) ∈ I(V) iff f(c, c) = 0 and

• unitary: ϕ(v) = τf(v, c), where \overline{τ} = −τ;

• symplectic: ϕ(v) = τf(v, c);

• orthogonal: p = 2 and ϕ(v) = \frac{f(v,c)}{q(c)}, where f is the polarization of the quadratic form q.

Since perpendicularity is a symmetric relations, the graph Γ(U) associated to any transvections subset U of I(V) is never one-way.

By the above a root subgroup is completely characterized by the centre of the defining transvection.

Conjecture

Theorem 11 (Brown, Humphries 1985). Let G ≤ Sp_{n}(K) be generated by root subgroups corresponding to a subset U of V = K^{n}. Then the following are equivalent:

1. G acts transitively on V^{#};

2. G = Sp_{n}(K);

3. V = ⟨U⟩ and Γ(U) is connected.

Given a transvection set U we have evidence for the following Conjecture 4. a) If \underline{L}(Ũ) = \overline{L}(U) and Γ(Ũ) is not one-way then G = Sp_{n}(q) ⋊ N.

b) If \underline{L}(Ũ) ≠ \overline{L}(U), then |\overline{L}(U) : \underline{L}(Ũ)| = 2, Γ(Ũ) is not one-way and G = SU_{n}(q) ⋊ N.

Some configuration

Theorem 12. Assume r, s, t are transvections with arcs (r, s), (s, t), (t, r), and, possibly, (s, r). Then up to conjugation in GL(n, K), (r, s, t) = (T_{31}(ξ)T_{32}(η), T_{12}(1), T_{23}(1)). Let F be the field generated by ξ, η and H = ⟨r, s, t⟩. Then H = SL(3, F) unless p = 2, ξ = 1, and ⟨η⟩ = 𝔽_{4}^{∗}. where H ≃ 3.A_{6}. In any case the linear span of x − 1 where x ranges over all transvections in opposite to t is 𝔰𝔩(3, F).

Diameter 2

Let Ũ be the G-conjugate closure of U. Proposition 5. Assume Γ(U) is connected, s, t ∈ Ũ. Then either s, t are opposite or there exists u ∈ Ũ opposite to both s and t.

Mutual Positions

We denote with T_{ij}(ξ) the elementary transvection 1 + ξe_{i}e_{j}^{t}, where e_{i} is the i-th standard vector. Lemma 6. Any pair of transvections (s, t) in SL(n, K) is conjugated in GL(n, K) : 2 to

1. (T_{12}(ξ), T_{21}(η));

2. (T_{12}(ξ), T_{34}(η));

3. (T_{12}(ξ), T_{13}(η)). Lemma 7. 1. r^{s} ∗ t = r ∗ t + (r ∗ s)(s ∗ t);

2. r ∗ t^{s} = r ∗ t − (r ∗ s)(s ∗ t).