Involutions in finite simple groups as products of conjugates

Let $G$ be a finite non-abelian simple group, $C$ a non-identity conjugacy class of $G$, and $\Gamma_C$ the Cayley graph of $G$ based on $C \cup C^{-1}$. Our main result shows that in any such graph, there is an involution at bounded distance from the identity.


Introduction
For a finite group G with a generating set S that is closed under inversion, the Cayley graph Γ(G, S) is defined to be the graph with vertex set G and edge set consisting of all {g, gs} for g ∈ G, s ∈ S.This paper is a contribution to the study of the Cayley graphs of finite (non-abelian) simple groups G in the case where the generating set is of the form C ∪ C −1 , where C is a non-identity conjugacy class of G.By [12], the diameter of such a Cayley graph is bounded above by c log |G|/ log |C|, where c is an absolute constant.Also, for G a simple group of Lie type of rank l, an explicit upper bound, linear in l, for the diameters of all such Cayley graphs, is obtained in [8].
Here we study a more refined question concerning distances in these Cayley graphs Γ C := Γ(G, C ∪ C −1 ) for simple groups G.In other words, for every non-identity conjugacy class C of G, we have (C ∪ C −1 ) k ∩ H = ∅ for some k ≤ d(H).
Clearly, there are some subsets H for which d(H) is equal to d G := max C =1 diam(Γ C ), hence grows with the rank of G. Our main results show that this is not the case when H = Inv(G), the set of involutions in the simple group G. Indeed, we show that d(Inv(G)) is bounded above by an absolute constant: explicitly, by 2 for alternating groups G = A n (n ≥ 6), by 3 for G = A 5 , and by 12 for classical groups.Constants for G an exceptional group of Lie type and G sporadic already follow from existing results: see Proposition 1.6.Thus for any non-identity conjugacy class C of a finite simple group G, there is a bounded product of elements of C ∪ C −1 that is an involution.
We now state our results.For classical groups, our proofs are inductive, which necessitates consideration of the general linear, unitary and orthogonal groups rather than just the simple groups.For ǫ = ±, GL ǫ n (q) denotes GL n (q) if ǫ = +1, and GU n (q) if ǫ = −1, with similar notation for SL ǫ n (q).Also when n is even, O ǫ n (q) denotes the orthogonal group O ± n (q) as usual, and when n is odd it denotes the orthogonal group O n (q).Theorem 1.1.Let G = GL ǫ n (q) and S = SL ǫ n (q), with n ≥ 2 and (n, q, ǫ) = (2, 2, ǫ) or (3, 2, −).Let g ∈ G \ Z(G), and C = g S .Then there exists an involution t ∈ S, Date: February 14, 2024. 1 with t ∈ Z(S) when n = 2, and elements , and C = g G .Then there exists an involution t ∈ G \ Z(G), and elements and S = Ω ǫ n (q), with n ≥ 7, and q odd if n is odd.Let g ∈ G \ Z(G), and C = g S .Then there exists an involution t ∈ S \ Z(S), and elements The following corollary follows directly from these theorems, except in the case where G = P SL 2 (q) with q odd, in which case C 3 = G for all non-identity conjugacy classes C, by [ In Section 5 we provide a different proof that for G = P SL(n, q) and 1 = g ∈ G, there is a product of at most 48 conjugates of g ±1 that is equal to an involution (see Theorem 5.1).Although this bound 48 is worse than the bound in Corollary 1.4, the proof is constructive, in the sense that it provides an algorithm that, given g, provides an explicit product of conjugates of g that is an involution.
For alternating groups we prove the following.
For the remaining simple groups, we note the following result, which follows directly from [8, Thm.2] (for exceptional groups of Lie type, including the Tits group 2 F 4 (2) ′ ) and [19] (for sporadic groups).Proposition 1.6.If G is a simple group of exceptional Lie type, then d(Inv(G)) ≤ 376; and if G is a sporadic simple group, then d(Inv(G)) ≤ 6.
Finally, we mention a consequence of our bounds on d(Inv(G)) for simple groups G. Namely, we obtain a new bound on the orbital diameters of primitive permutation groups of simple diagonal type, improving a result in [15].See Section 7 for details.
The layout of the paper is as follows.In Sections 2-4 we prove Theorems 1.1-1.3,and Section 5 contains our constructive proof for P SL n (q).The proof of Theorem 1.5 can be found in Section 6, and the final section contains our result on orbital diameters.

Proof of Theorem 1.1
The proof will be inductive, and the following lemma is the n = 2 case, which forms the base for the induction.Lemma 2.1.Let G = GL ǫ 2 (q) and S = SL 2 (q), with q > 2. If g ∈ G \ Z(G), and C = g S , then there exist elements Proof First choose h ∈ S such that [g, h] = I, and note that [g, h] ∈ CC −1 ⊆ S. If q is even, then (CC −1 ) 3 = S by [1, Thm 4.2 (a)], and the conclusion follows.
(1) Consider first the case where g is decomposable, i.e. g lies in a proper subgroup Assume that there exists i such that g i ∈ Z(GL(V i )), say i = 1.Then by induction, one of the following holds: In case (i), if we let x j = x ′ j ⊕ g 2 and y j = y ′ j ⊕ g −1 2 , then these are S-conjugates of g and g −1 , with product t 1 ⊕ 1, a non-central involution in S. In case (ii) we have n ≥ 5 by assumption, so n 2 ≥ 3.If g 2 = 1, work as above with g 2 instead of g 1 ; and if g 2 = 1, we can replace g 1 by (g 1 , 1) ∈ GL 3 (2) and proceed as above.Now assume that g i ∈ Z(GL(V i )) for i = 1, 2, so g = λI n 1 ⊕ µI n 2 for some λ, µ ∈ F * q .Then we can replace g by the S-conjugate h 1 ⊕h 2 , where h 1 = (λI n 1 −1 , µ), h 2 = (λ, µI n 2 −1 ), and proceed as in the previous paragraph.
(2) Now consider the case where g is indecomposable, i.e. does not preserve any proper decomposition V = V 1 ⊕ V 2 .Let g = su be the Jordan decomposition of g, where s is semisimple, u is unipotent, and us = su.The structure of the centralizer C G (s) is well known, and we have ) with md = n, and also u is a single Jordan block J m ∈ GL m (q d ).Moreover, if 2 < m < n, or if m = 2 with q even, then we can apply induction in the group GL m (q d ) to obtain the conclusion of the theorem, so assume that m = 1, 2 or n (with q odd if m = 2).
If m = 1 then g = s and C G (g) = GL 1 (q n ); if m = n then g = λJ with J a unipotent Jordan block, and C G (g) has order (q − 1)q n−1 ; and if Suppose first that m = n, so g = λJ.Let C = J S , the conjugacy class of J in S = SL n (q), and let t be a non-central involution in S. By [1, p.43], the number of solutions to the equation 6  1 To show that this sum is nonzero, we use the following facts; in (c), k(S) denotes the number of conjugacy classes in S: This is less than 1 when n ≥ 4, so the sum in ( 2) is nonzero, and the conclusion follows in this case.For n = 3, the character table of SL 3 (q) is available in [14].We can replace the bound for |χ(J)| = |χ(J −1 )| by 2q + 1 for all χ ∈ Irr(S) \ 1, so the sum in ( 2) is nonzero for q ≥ 7 and the conclusion follows.Now suppose that m = 1, so g = s and C G (g) = GL 1 (q n ).The determinant map C G (g) → F * q is surjective, so we have g G = g S .Hence, as above, it is sufficient to prove that N ′ > 0, where The contribution of the q − 1 linear characters of G to N ′ is q − 1.For the nonlinear characters χ ∈ Irr(G), again using [5,18], we have Hence the contribution of the nonlinear characters to N ′ is of absolute value less than q n (q n/2 ) 12 q 10(n−1) = q −3n+10 , which is less than 1 for n ≥ 4. For n = 3, the result follows as again by [17] we can replace the bound for |χ(g)| = |χ(g −1 )| by 6 for all nonlinear χ ∈ Irr(G).Finally, suppose m = 2 with q odd.Here The argument of the previous paragraph goes through with G * replacing G, and replacing the inequality in (c) by k(G * ) ≤ 1 2 (q n + 3q n−1 ) (see [5,Cor. 3.7]).This completes the proof of the theorem for G = GL n (q).We now indicate the changes in the proof needed for G = GU n (q).First, we need to deal computationally with some small cases, namely n = 3 (q ≤ 7), n = 4 (q ≤ 3), and (n, q) = (5, 2), (6,2).( 4) We now assume none of these cases holds.If g is decomposable, i.e. g lies in a proper subgroup then we argue by induction in similar fashion to the GL n (q) case, as follows.Write g = g 1 ⊕g 2 with g i ∈ GU (V i ), and assume first that there exists i such that g i ∈ Z(GU (V i )), say i = 1.Then by induction, one of the following holds: (i) there is an involution t 1 ∈ SU (V 1 ), and SU (V 1 )-conjugates x ′ j and y ′ j (1 ≤ j ≤ 6) of g 1 and g −1 1 respectively, such that 6 In case (i), if we let x j = x ′ j ⊕ g 2 and y j = y ′ j ⊕ g −1 2 , then these are S-conjugates of g and g −1 , with product t 1 ⊕ 1, a non-central involution in S. In case (ii) we have n ≥ 7 by assumption, so n 2 ≥ 4. If g 2 ∈ Z(GU (V 2 )), work as above with g 2 instead of g 1 ; otherwise, g 2 = ωI n 2 , we can replace g 1 by (g 1 , ω) ∈ GU 4 (2) and proceed as above.
So assume g is indecomposable, and let g = su be the Jordan decomposition of g.Then where 2m i d i + r i e i = n and all e i are odd.Since g is indecomposable, k + l = 1.As in the GL case, we can apply induction to see that one of the following holds: (i) C G (s) = GU n (q), and g = λJ where J is a single Jordan block in G; , where q is odd, and u is a single Jordan block The maximal possible value of |C G (g)| is its value in (i), namely (q + 1)q n−1 .Consider first case (i).We need to argue as in the GL case that the sum in ( 2) is nonzero.To do this we use the facts: • χ(1) ≥ q n −q q+1 for all χ ∈ Irr(S) \ 1, by [18, Thm.|χ(J) 6 χ(J −1 ) 6 | χ(1) 10 ≤ 8.26q n−1 ((q + 1)q n−1 ) 6 ((q n − q)/(q + 1)) 10 .
In case (ii) we have g S = g G and as before we need to argue that the sum in (3) is positive.For this we argue as in the previous paragraph, using the bound k(G) ≤ 8.26q n from [5, Prop.3.9], for n = 3 also the bound |χ(g)| = |χ(g −1 )| ≤ 6 from [4], and for (n, q) = (4, 4) using the fact that k(G) = 470 from a computation in GAP.
Finally, in case (iii) as in the GL case we have g S = g G * , where G * is of index 2 in G, and we argue in the usual way, using the bound k(G * ) ≤ 4.13q n−1 (q + 1) from [5,Cor. 3.11] (and also for (n, q) = (5, 2) and (4, 4), the exact values of k(G * ), computed in GAP).
This completes the proof of Theorem 1.1.

Proof of Theorem 1.2
We prove by induction on m the statement of Theorem 1.2, but including also the case where m = 1 (which is already proved in Lemma 2.1).So let G = Sp 2m (q) = Sp(V ) with m ≥ 2, let g ∈ G \ Z(G) and C = g G .Our approach is similar to that in the previous section.
(1) Consider first the case where g is decomposable, i.e. g lies in a proper subgroup Sp 2m 1 (q) × Sp 2m 2 (q) of G preserving an orthogonal decomposition V = V 1 ⊕ V 2 , where dim V i = 2m i .Say g = g 1 ⊕ g 2 , where g i ∈ Sp(V i ).Then induction gives the conclusion in exactly the same way as in (1) of the previous section.
(2) Now suppose g is indecomposable, and let g = su be the Jordan decomposition.The structure of C G (s) is as follows: if for ǫ ∈ {±1}, V ǫ denotes the ǫ-eigenspace of s, then V ǫ is non-degenerate and Sp 2s i e i (q) ≤ G. Since g is indecomposable, there is only one factor in C G (s), and so one of the following holds: In case (i), we use [7, Prop.2.3, Thm.3.1] for the classification of unipotent classes in G.If q is odd, the indecomposable such classes are those labelled V β (2m) (a single Jordan block) and W (m) for m odd (two blocks of size m); for the latter class, g = ±W (m) lies in a subgroup GL m (q) of G, and the conclusion follows from Theorem 1.1.If q is even, the indecomposable unipotent classes are those labelled V β (2m), W (m) and W α (m) (m odd); elements in the classes W (m) and W α (m) lie in subgroups GL m (q) and GU m (q) respectively, so again the result follows from Theorem 1.1 in these cases.In summary, in case (i) we may take it that g = ±V β (2m), a single Jordan block.The centralizer order is |C G (g)| = 2q m (see [11,Chap. 7]).
In case (ii), u is a single Jordan block J r ∈ C G (s) = GL ± r (q d ) with rd = m.The maximal centralizer order occurs when C G (s) = GU m (q), and is (q + 1)q m−1 .We conclude that in both cases (i) and (ii), At this point we use character theory, as in the previous section.Again, we need to show that the sum in (3) is positive.
Once again we check that this is less than 1 for m ≥ 2 and q even, with (m, q) = (2, 2) or (3,2) for which cases the result follows by calculations in GAP.
This completes the proof of Theorem 1.2.

Proof of Theorem 1.3
This is similar to the previous section, but requires a bit more care in some places.For the induction argument we shall need the following lemma.Lemma 4.1.Let G = O ǫ 4 (q) and S = Ω ǫ 4 (q), with q > 3. Let g ∈ G \ Z(G), and C = g S .Then there exists a non-central involution t ∈ S, and elements Proof We have O ǫ 4 (q) ∼ = Ω ǫ 4 (q).2 a , where a = (2, q − 1), and Ω − 4 (q) ∼ = P SL 2 (q 2 ), Ω + 4 (q) ∼ = SL 2 (q) • SL 2 (q).Choose x ∈ S such that h := [g, x] ∈ S \ Z(S).It is sufficient to find 6 conjugates of h with product equal to a non-central involution in S.This can be done, as was shown in the proof of Lemma 2.1.✷ We now prove Theorem 1.3.Let G = O ǫ n (q) and S = Ω ǫ n (q), with n ≥ 5, and q odd if n is odd.Let g ∈ G \ Z(G), and C = g S .We assume that (n, q) = (7, 3), (8, 2), (8,3).(7) These cases are easily handled by computation.As usual we proceed by induction on n.The cases n = 5 and n = 6 are covered by Theorems 1.1 and 1.2 in view of the isomorphisms Ω 5 (q) ∼ = P Sp 4 (q) and P Ω ǫ 6 (q) ∼ = P SL ǫ 4 (q).So assume n ≥ 7.
(1) Consider first the case where g is decomposable, i.e. g lies in a proper subgroup where g i ∈ O(V i ).We need to be a little careful in applying induction in this case.Let ).If n 1 ≥ 5, we can apply induction to find 12 conjugates of g ±1 1 with product a non-central involution in Ω(V 1 ), giving the conclusion in the usual way.If n 1 = 4, then q > 3 by the exclusions in (7), and so the conclusion follows in the same way, using Lemma 4.1.Now suppose g 1 ∈ Z(O(V 1 )), so g 1 = δI n 1 with δ ∈ {1, −1}.Take any proper orthogonal decomposition and g 2 by δI W 1 , the argument of the previous paragraph goes through.
We check that this is less than 1 unless (n, q) as in the exclusions (7), and the conclusion of the theorem follows in the usual way.This completes the proof of Theorem 1.3.

A constructive algorithm
In this section we give an alternative proof to the case ǫ = +1 of Theorem 1.1.Although it requires a larger number of conjugates of g ±1 , the conjugating elements are explicitly determined up to possibly passing to generalized Jordan form, for which algorithms are well-known (see for instance [16]).One might be able to apply the method to other groups of Lie type, although with considerably more work.and note that sg Then for we get and putting everything together we obtain in the end reducing the problem to the case n = 2.Note that in the case n = 2 we obtained the conclusion for h(1) with k = 3, so it follows here for g with k = 12.
In case (2), n must be even.Assume n ≥ 8, take s(y) as in the previous case and Then reducing again to n = 2; as above, this gives the conclusion with k = 12.For n = 6, use the same procedure replacing For n = 4, take instead reducing again to n = 2; in this n = 2 case we only obtained the conclusion with k = 12, so it follows for g with k = 24.

A consequence on orbital diameters
Let G be a finite group acting transitively on set Ω.The orbitals of G are its orbits on Ω × Ω, and the diagonal orbital is {(α, α) : α ∈ Ω}.For a non-diagonal orbital Γ, we define the corresponding orbital graph to be the undirected graph with vertex set Ω and edge set {{α, β} : (α, β) ∈ Γ}.By [2, Thm.3.2A], the orbital graphs are all connected if and only if G acts primitively on Ω, in which case the orbital diameter of G is defined to be the supremum of the diameters of its orbital graphs (see [9]).Let us denote this by orbdiam(G).
In [15], a study is made of the orbital diameters of primitive groups of simple diagonal type.We are able to use our results to improve one of the theorems in that paper, namely [15, Thm.6.1], as follows.Let T be a non-abelian simple group, k ≥ 2 an integer, and let S(T, k) := T k .S k denote the semidirect product in which S k permutes the coordinates in T k naturally.Define D = {(t, . . ., t) : t ∈ T }, a diagonal subgroup of T k , and let Ω = (T k : D) be the set of right cosets of D in T k .Then S(T, k) acts primitively on Ω, where T k acts by right multiplication, and S k by permuting the components of coset representatives, and this is a primitive group of simple diagonal type (see [10]).
The following result determines the orbital diameter of S(T, k) up to a multiplicative constant.Recall our definition d T := max C =1 diam(Γ C ) from Section 1. Proof The lower bound is given by [15,Thm. 3.1].An upper bound of 24(k − 1)d 2 T is proved in [15,Thm. 6.1].However, in the proof of that result, it is necessary to construct an involution as a product of length d T of conjugates of some element, which accounts for one of the factors d T in the upper bound.This factor can therefore be replaced by C, giving the required upper bound.✷ Note that d T grows linearly in the rank of T for groups of Lie type, and linearly in the degree n of T = A n , whereas C is bounded absolutely in all cases.The previous upper bound [15,Thm. 6.1] was quadratic in d T .
For a subset H ⊆ G not containing the identity, define a number d(H) as follows.Let d C denote the distance function on Γ C , and set d C (H) = min (d C (1, h) : h ∈ H) , so that d C (H) = min k : (C ∪ C −1 ) k ∩ H = ∅ .Now define d(H) = max C =1 d C (H).