The Ore Conjecture

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The Ore Conjecture

Eamonn O’Brien

University of Auckland

December 2010

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Commutators

G finite group

G⁰ = h[x , y ] : x , y ∈ G i is the subgroup generated by commutators

Not every g ∈ G⁰ is a commutator [x , y ].

Group H of order 96, |H⁰| = 32 and contains 29 commutators.

Eamonn O’Brien The Ore Conjecture

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Commutators

G finite group

G⁰ = h[x , y ] : x , y ∈ G i is the subgroup generated by commutators Not every g ∈ G⁰ is a commutator [x , y ].

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Commutators

G finite group

G⁰ = h[x , y ] : x , y ∈ G i is the subgroup generated by commutators Not every g ∈ G⁰ is a commutator [x , y ].

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But every element g of G⁰ is a product of commutators.

Problem

Can we bound the length of such a product independently of g ?

Theorem (Nikolov & Segal, 2007)

There exists a function f such that if G is a d -generator finite group, then every element of G⁰ is a product of f (d ) commutators. Special interest: G finite simple group.

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Problem

There exists a function f such that if G is a d -generator finite group, then every element of G⁰ is a product of f (d ) commutators. Special interest: G finite simple group.

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Problem

There exists a function f such that if G is a d -generator finite group, then every element of G⁰ is a product of f (d ) commutators.

Special interest: G finite simple group.

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Problem

There exists a function f such that if G is a d -generator finite group, then every element of G⁰ is a product of f (d ) commutators.

Special interest: G finite simple group.

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The Ore Conjecture (1951)

Every element of a finite simple group is a commutator.

Ore proved it for A_n: case by case, every relevant combination of cycles dealt with in turn.

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The Ore Conjecture (1951)

Every element of a finite simple group is a commutator.

Ore proved it for A_n: case by case, every relevant combination of cycles dealt with in turn.

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The LOST result

Liebeck, O’B, Shalev, Tiep (JEMS, 2010) Theorem

If G is a finite non-abelian simple group, then every g ∈ G is a commutator.

In fact: every element of every quasisimple classical group is a commutator.

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The LOST result

Liebeck, O’B, Shalev, Tiep (JEMS, 2010) Theorem

If G is a finite non-abelian simple group, then every g ∈ G is a commutator.

In fact: every element of every quasisimple classical group is a commutator.

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Not true for arbitrary quasi-simple groups: no element of order 12 in 3A6 is a commutator.

Theorem

The only quasisimple groups with non-central elements which are not commutators are covers of A6, A7, L3(4) and U4(3).

Corollary

Every element of every finite quasisimple group is a product of two commutators.

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Not true for arbitrary quasi-simple groups: no element of order 12 in 3A6 is a commutator.

Theorem

The only quasisimple groups with non-central elements which are not commutators are covers of A6, A7, L3(4) and U4(3).

Corollary

Every element of every finite quasisimple group is a product of two commutators.

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Overview of the lecture

A broader context

The basic approach

A sketch of the proof

Waring type problems

Shalev et al.: program to express group elements as short products of values of fixed word w .

Let w = w (x1, . . . , xd) be element of free group Fd on x1, . . . , xd. Consider word map

wG : G^d 7−→ G

(g1, . . . , gd) 7−→ w (g₁, . . . , gd) Set of all group elements w (g₁, . . . , g_d) is W (G ).

How large is W (G )? Jones (1974) showed it’s non-trivial for all w 6= 1 if G is large enough.

Can we express g ∈ G as short product of elements of W (G )? Waring: express every integer as a sum of f (k) k-th powers.

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Waring type problems

w_G : G^d 7−→ G

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Waring type problems

w_G : G^d 7−→ G

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Waring type problems

w_G : G^d 7−→ G

Can we express g ∈ G as short product of elements of W (G )?

Waring: express every integer as a sum of f (k) k-th powers.

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Waring type problems

w_G : G^d 7−→ G

Can we express g ∈ G as short product of elements of W (G )?

Waring: express every integer as a sum of f (k) k-th powers.

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Other much studied words: x₁^k in Burnside-type problems, x^py^p where p is prime.

Theorem (Shalev, 2009)

For each w 6= 1, there exists N = Nw depending only on w such that if G is a finite simple group of order at least N then W (G )³ = G .

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Other much studied words: x₁^k in Burnside-type problems, x^py^p where p is prime.

For each w 6= 1, there exists N = N_w depending only on w such that if G is a finite simple group of order at least N then W (G )³ = G .

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Covering numbers

G finite simple group, C 6= {1} is a conjugacy class.

Then there exists k ∈ P such that C^k = G .

Minimal such k over all classes C is covering number c(G ) of G . Ellers, Gordeev & Herzog (1999); Lawther & Liebeck (1998) Theorem

c(An) = d(n − 1)/2e

c(Gr(q)) ≤ mr for some absolute constant m.

Theorem (Liebeck & Shalev, 2001) c(C , G ) ≤ m log |G |/log |C |

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Covering numbers

c(An) = d(n − 1)/2e

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Covering numbers

Minimal such k over all classes C is covering number c(G ) of G .

Ellers, Gordeev & Herzog (1999); Lawther & Liebeck (1998) Theorem

c(An) = d(n − 1)/2e

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Covering numbers

c(A_n) = d(n − 1)/2e

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Covering numbers

c(A_n) = d(n − 1)/2e

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Thompson’s conjecture (1985)

Every finite non-abelian simple group G contains a conjugacy class C with C²= G .

Lemma

Thompson implies Ore.

Proof.

Let C = x^G. Now 1 ∈ G = C² so x⁻¹∈ C and G = (x⁻¹)^Gx^G. Hence every element of G is a commutator.

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Thompson’s conjecture (1985)

Lemma

Proof.

Let C = x^G. Now 1 ∈ G = C² so x⁻¹∈ C and G = (x⁻¹)^Gx^G. Hence every element of G is a commutator.

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Thompson’s conjecture (1985)

Lemma

Proof.

Let C = x^G. Now 1 ∈ G = C² so x⁻¹ ∈ C and G = (x⁻¹)^Gx^G. Hence every element of G is a commutator.

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Related probabilistic work

Shalev (2009): if g is a random element of finite simple group G , then the probability that g is a commutator tends to 1 as

|G | 7→ ∞.

If G = G_q(r ), a Lie type simple group of rank r over field of size q, then probability is at least 1 − cq^−2r where c is absolute constant. Garion & Shalev (2009): For finite simple group G , the map α : G × G 7−→ G defined by α(x , y ) = [x , y ] is almost equidistributed, so almost all elements are commutators. Applications to the product replacement algorithm. Theorem (Shalev, 2009)

There exists an absolute constant c such that every finite simple group G of order at least c has a conjugacy class C such that C² = G . If x ∈ G is random, then probability that (x^G)³ = G tends to 1 as |G | 7→ ∞.

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Related probabilistic work

|G | 7→ ∞.

If G = G_q(r ), a Lie type simple group of rank r over field of size q, then probability is at least 1 − cq^−2r where c is absolute constant.

Garion & Shalev (2009): For finite simple group G , the map α : G × G 7−→ G defined by α(x , y ) = [x , y ] is almost equidistributed, so almost all elements are commutators. Applications to the product replacement algorithm. Theorem (Shalev, 2009)

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Related probabilistic work

|G | 7→ ∞.

Garion & Shalev (2009): For finite simple group G , the map α : G × G 7−→ G defined by α(x , y ) = [x , y ] is almost equidistributed, so almost all elements are commutators.

Applications to the product replacement algorithm. Theorem (Shalev, 2009)

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Related probabilistic work

|G | 7→ ∞.

Garion & Shalev (2009): For finite simple group G , the map α : G × G 7−→ G defined by α(x , y ) = [x , y ] is almost equidistributed, so almost all elements are commutators.

Applications to the product replacement algorithm.

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The Thompson criterion

Theorem (Frobenius, 1896)

Let G be a finite group, let g be a fixed element of G , and for 1 ≤ i ≤ t let C_i be a conjugacy class in G with representative x_i. The number of solutions to the equationQt

i =1yi = g with yi ∈ C_i is equal to

|C₁| · · · |C_t|

|G |

X

χ∈Irr(G )

χ(x₁) · · · χ(x_t)χ(g⁻¹) χ(1)^t−1 ,

where Irr(G ) is the set of ordinary irreducible characters of G . Hence g ∈ C² if and only if

X χ(C )²χ(g⁻¹) χ(1) 6= 0

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The Ore criterion

Theorem (Frobenius, 1896) For fixed g ∈ G ,

#{(x , y ) ∈ G × G | g = [x , y ]} = |G | X

χ∈Irr(G)

χ(g ) χ(1)

To show g ∈ G is commutator, suffices to show that X

χ∈Irr(G)

χ(g ) χ(1) 6= 0 Or

| X

χ(1)>1

χ(g ) χ(1)| < 1

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The Ore criterion

#{(x , y ) ∈ G × G | g = [x , y ]} = |G | X

χ∈Irr(G)

χ(g ) χ(1)

To show g ∈ G is commutator, suffices to show that

X

χ∈Irr(G)

χ(g ) χ(1) 6= 0 Or

| X

χ(1)>1

χ(g ) χ(1)| < 1

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The Ore criterion

#{(x , y ) ∈ G × G | g = [x , y ]} = |G | X

χ∈Irr(G)

χ(g ) χ(1)

χ∈Irr(G)

χ(g ) χ(1) 6= 0

Or

| X

χ(1)>1

χ(g ) χ(1)| < 1

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The Ore criterion

#{(x , y ) ∈ G × G | g = [x , y ]} = |G | X

χ∈Irr(G)

χ(g ) χ(1)

χ∈Irr(G)

χ(g ) χ(1) 6= 0 Or

| X

χ(1)>1

χ(g ) χ(1)| < 1

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The key step

X

χ∈Irr(G)

|χ(g )|² = |C_G(g )|

Partition elements of G by centraliser size

If G a finite simple group and g ∈ G has small centraliser then main contribution to

|G | X

χ∈Irr(G)

χ(g ) χ(1) comes from the trivial character χ = 1.

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The key step

X

χ∈Irr(G)

|χ(g )|² = |C_G(g )|

|G | X

χ∈Irr(G)

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The key step

X

χ∈Irr(G)

|χ(g )|² = |C_G(g )|

|G | X

χ∈Irr(G)

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Shalev’s probabilistic results

If g ∈ G has small centraliser, then

#{(x , y ) ∈ G × G | g = [x , y ]} = |G |(1 + o(1)) where o(1) 7→ 0 as |G | 7→ ∞ and g is a commutator when G is large enough.

So elements with small centralisers are commutators. Almost all elements of G have small centralisers.

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Shalev’s probabilistic results

So elements with small centralisers are commutators.

Almost all elements of G have small centralisers.

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Shalev’s probabilistic results

So elements with small centralisers are commutators.

Almost all elements of G have small centralisers.

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Earlier work on Thompson / Ore

Ore (1951): conjectured and proved Ore for A_n.

Hs¨u (1965): Thompson for A_n.

R.C. Thompson (1962-63): Ore for PSL_n(q). Use structure of G to write g = [x , y ] based on various kinds of factorisations.

Use similarity of matrices.

Brenner (1983), Sourour (1986), Lev (1994): Thompson for PSLn(q).

Neub¨user, Pahlings, Cleuvers (1988): sporadics.

Gow (1988): PSp_n(q), q ≡ 1 mod 4.

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Bonten (1993): G Lie type, rank r . There exists a constant q0 such that every element of Gr(q) is a commutator for q > q₀. Exploited Frobenius and character ratios to obtain result for exceptionals of rank at most 4.

Gow (2000): If C is a class of regular semisimple real elements in simple group of Lie type, then C²= G .

Theorem (Ellers & Gordeev, 1998)

If Chevellay group G has two regular semisimple elements h₁ and h2 in a maximal split torus, then G \ Z (G ) ⊂ C1C2.

Ore follows if G has regular semisimple element h in maximal split torus; Thompson if h is real.

Ore and Thompson hold for finite simple groups if q ≥ 8.

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If Chevellay group G has two regular semisimple elements h1 and h2 in a maximal split torus, then G \ Z (G ) ⊂ C1C2.

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Sketch of LOST proof

To show g ∈ G is commutator, suffices to show that

X

χ∈Irr(G)

χ(g ) χ(1) 6= 0 or

| X

χ(1)>1

χ(g ) χ(1)| < 1 Key: partition elements by centraliser size.

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Sketch of LOST proof

χ∈Irr(G)

χ(g ) χ(1) 6= 0

or

| X

χ(1)>1

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Sketch of LOST proof

χ∈Irr(G)

χ(g ) χ(1) 6= 0 or

| X

χ(1)>1

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|C

_G

(g )| is small

Use existing knowledge of chars, Deligne-Lusztig theory, and the theory of dual pairs and Weil characters of classical groups to construct explicitly irreducible characters of relatively small degrees, and to derive information on their character values.

Show |χ(g )|/χ(1) is small for χ 6= 1, so main contribution to P

χ∈ Irr(G)χ(g )/χ(1) comes from χ = 1.

Hence deduce that sum is positive, and so elements with small centralisers are commutators.

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|C

_G

(g )| is small

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|C

_G

(g )| is small

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|C

_G

(g )| is large

Reduce problem to groups of smaller rank and use induction. Usually such g ∈ G has decomposition into Jordan blocks, and so lies in direct product of smaller classical groups.

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|C

_G

(g )| is large

Reduce problem to groups of smaller rank and use induction.

Usually such g ∈ G has decomposition into Jordan blocks, and so lies in direct product of smaller classical groups.

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|C

_G

(g )| is large

Reduce problem to groups of smaller rank and use induction.

Usually such g ∈ G has decomposition into Jordan blocks, and so lies in direct product of smaller classical groups.

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Breakable elements

Let G = Cl (V ) = Sp(V ), SU(V ) or Ω(V ).

Definition

x ∈ G is breakable if there is a proper, nonzero, non-degenerate subspace W of V such that x = (x₁, x₂) ∈ Cl (W ) × Cl (W^⊥), and one of the following holds:

both factors Cl (W ) and Cl (W^⊥) are perfect groups;

Cl (W ) is perfect, and x₂ is a commutator in Cl (W^⊥).

Otherwise, x is unbreakable.

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Lemma

Suppose that whenever W is a non-degenerate subspace of V such that Cl (W ) is a perfect group, every unbreakable element of Cl (W ) is a commutator in Cl (W ). Then every element of the perfect group G is a commutator.

Proof.

The proof goes by induction on dim V .

The inductive hypothesis holds for all perfect subgroups of G of the form Cl (X ) with X a non-degenerate subspace of V . If x ∈ G is unbreakable, then it is a commutator by hypothesis. Otherwise x is breakable, so x = (x₁, x₂) ∈ Cl (W ) × Cl (W^⊥) satisfies (1) or (2).

In either case, by induction x1, x2 are commutators in Cl (W ), Cl (W^⊥) respectively, and so x is a commutator, as required.

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Lemma

Proof.

The inductive hypothesis holds for all perfect subgroups of G of the form Cl (X ) with X a non-degenerate subspace of V . If x ∈ G is unbreakable, then it is a commutator by hypothesis. Otherwise x is breakable, so x = (x₁, x₂) ∈ Cl (W ) × Cl (W^⊥) satisfies (1) or (2).

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Lemma

Proof.

The inductive hypothesis holds for all perfect subgroups of G of the form Cl (X ) with X a non-degenerate subspace of V .

If x ∈ G is unbreakable, then it is a commutator by hypothesis. Otherwise x is breakable, so x = (x₁, x₂) ∈ Cl (W ) × Cl (W^⊥) satisfies (1) or (2).

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Lemma

Proof.

The inductive hypothesis holds for all perfect subgroups of G of the form Cl (X ) with X a non-degenerate subspace of V . If x ∈ G is unbreakable, then it is a commutator by hypothesis.

Otherwise x is breakable, so x = (x₁, x₂) ∈ Cl (W ) × Cl (W^⊥) satisfies (1) or (2).

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Lemma

Proof.

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Lemma

Proof.

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Difficulties with reduction

Some blocks may lie in a group which is not perfect, such as Sp2(2), Sp2(3), Sp4(2), Ω⁺₄(2); or in orthogonal case blocks may have determinant −1.

Unitary groups: Jordan blocks can have many different determinants. e.g. 8 possible values for PSU_n(7).

Instead solve certain equations in unitary groups, and establish certain properties of unitary matrices in small dimensions.

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Difficulties with reduction

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Difficulties with reduction

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Difficulties with reduction

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Proving Ore for unbreakable elements

Enough to prove that unbreakable g ∈ G = Cl (V ) is commutator.

If g unbreakable, then |CG(g )| is small.

For unbreakable g and n > n₀, prove that g is a commutator.

Induction base: prove Ore for Cl_n(q) for n ≤ n₀.

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Proving Ore for unbreakable elements

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Proving Ore for unbreakable elements

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Sp

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Lemma

Assume n ≥ 7, and let x be an unbreakable element of G = Sp(V ) = Sp_2n(2). Then |C_G(x )| < 2²ⁿ⁺¹⁵.

Based on detailed analysis of Jordan forms of elements.

Let k(G ) be number of conjugacy classes of G . Theorem (Fulman & Guralnick, 2009)

k(Sp2n(q)) ≤ 12qⁿ if q is odd, and k(Sp2n(q)) ≤ 17qⁿ if q is even.

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Sp

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Lemma

Assume n ≥ 7, and let x be an unbreakable element of G = Sp(V ) = Sp_2n(2). Then |C_G(x )| < 2²ⁿ⁺¹⁵.

Based on detailed analysis of Jordan forms of elements.

Let k(G ) be number of conjugacy classes of G . Theorem (Fulman & Guralnick, 2009)

k(Sp2n(q)) ≤ 12qⁿ if q is odd, and k(Sp2n(q)) ≤ 17qⁿ if q is even.

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Theorem (Guralnick & Tiep, 2004)

Let G = Sp2n(q) with q even, n ≥ 4. There is a collection W of q + 3 irreducible characters of G , such that

χ(1) ≥ ^(qⁿ^−1)(q_2(q+1)ⁿ^−q) if χ ∈ W,

χ(1) ≥ ¹₂(q²ⁿ− 1)(qⁿ⁻¹− 1)(qⁿ⁻¹− q²)/(q⁴− 1) for 1 6= χ ∈ Irr(G)\W.

Partition sum of non-trivial char values for unbreakable g ∈ G as S₁(g ) = X

χ∈W

χ(g )

χ(1), S2(g ) = X

16=χ∈ Irr(G)\W

χ(g ) χ(1), and show |S₁(g )| + |S₂(g )| < 1.

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Some facts

P

χ∈ Irr(G)|χ(g )| ≤ k(G )^1/2|C_G(g )|^1/2

If χ1, . . . , χ_k ∈ Irr(G) are distinct characters of degree ≥ N, then

X

χ∈ Irr(G), χ(1)≥N

|χ(g )|

χ(1) ≤ k(G )^1/2|C_G(g )|^1/2

N .

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Some facts

P

χ∈ Irr(G)|χ(g )| ≤ k(G )^1/2|C_G(g )|^1/2

If χ1, . . . , χ_k ∈ Irr(G) are distinct characters of degree ≥ N, then

X

χ∈ Irr(G), χ(1)≥N

|χ(g )|

χ(1) ≤ k(G )^1/2|C_G(g )|^1/2

N .

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We can readily bound S2(x ).

Lemma

Suppose n ≥ 7. If |C_G(x )| < 2²ⁿ⁺¹⁵, then |S₂(x )| < 0.6.

Proof.

S₂(x ) is sum over at most k(G ) characters, each of degree at least 1

30(2²ⁿ− 1)(2ⁿ⁻¹− 1)(2ⁿ⁻¹− 4). Deduce that

|S₂(x )| < 30√

17 · 2^n/2|C_G(x )|^1/2 (2²ⁿ− 1)(2ⁿ⁻¹− 1)(2ⁿ⁻¹− 4). This is less than 0.6 when |CG(x )| < 2²ⁿ⁺¹⁵ and n ≥ 7.

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Lemma

Proof.

30(2²ⁿ− 1)(2ⁿ⁻¹− 1)(2ⁿ⁻¹− 4).

Deduce that

|S₂(x )| < 30√

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Lemma

Proof.

30(2²ⁿ− 1)(2ⁿ⁻¹− 1)(2ⁿ⁻¹− 4).

Deduce that

|S₂(x )| < 30√

17 · 2^n/2|C_G(x )|^1/2 (2²ⁿ− 1)(2ⁿ⁻¹− 1)(2ⁿ⁻¹− 4).

This is less than 0.6 when |CG(x )| < 2²ⁿ⁺¹⁵ and n ≥ 7.

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Lemma

Proof.

30(2²ⁿ− 1)(2ⁿ⁻¹− 1)(2ⁿ⁻¹− 4).

Deduce that

|S₂(x )| < 30√

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Lemma

Suppose n ≥ 7. If |C_G(x )| < 2²ⁿ⁺¹⁵, then |S₁(x )| < 0.2.

Bound for S₁ based on a detailed analysis of the characters in W, taken from Guralnick & Tiep (2004).

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