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# 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 ].

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

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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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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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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.

Eamonn O’Brien The Ore Conjecture

(7)

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.

(8)

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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.

Eamonn O’Brien The Ore Conjecture

(9)

### The Ore Conjecture (1951)

Every element of a finite simple group is a commutator.

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

Eamonn O’Brien The Ore Conjecture

(11)

### 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.

Eamonn O’Brien The Ore Conjecture

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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.

Eamonn O’Brien The Ore Conjecture

(15)

### Overview of the lecture

 The basic approach

 A sketch of the proof

 Related questions

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### 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 : Gd 7−→ G

(g1, . . . , gd) 7−→ w (g1, . . . , gd) Set of all group elements w (g1, . . . , gd) 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.

Eamonn O’Brien The Ore Conjecture

(17)

### 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 : Gd 7−→ G

(g1, . . . , gd) 7−→ w (g1, . . . , gd) Set of all group elements w (g1, . . . , gd) 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.

(18)

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### 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 : Gd 7−→ G

(g1, . . . , gd) 7−→ w (g1, . . . , gd) Set of all group elements w (g1, . . . , gd) 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.

Eamonn O’Brien The Ore Conjecture

(19)

### 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 : Gd 7−→ G

(g1, . . . , gd) 7−→ w (g1, . . . , gd) Set of all group elements w (g1, . . . , gd) 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.

(20)

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### 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 : Gd 7−→ G

(g1, . . . , gd) 7−→ w (g1, . . . , gd) Set of all group elements w (g1, . . . , gd) 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.

Eamonn O’Brien The Ore Conjecture

(21)

Other much studied words: x1k in Burnside-type problems, xpyp 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: x1k in Burnside-type problems, xpyp 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 .

Eamonn O’Brien The Ore Conjecture

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### Covering numbers

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

Then there exists k ∈ P such that Ck = 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

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

Then there exists k ∈ P such that Ck = 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 |

Eamonn O’Brien The Ore Conjecture

(25)

### Covering numbers

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

Then there exists k ∈ P such that Ck = 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 |

(26)

logo

### Covering numbers

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

Then there exists k ∈ P such that Ck = 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 |

Eamonn O’Brien The Ore Conjecture

(27)

### Covering numbers

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

Then there exists k ∈ P such that Ck = 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 |

(28)

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

Eamonn O’Brien The Ore Conjecture

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

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

Eamonn O’Brien The Ore Conjecture

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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 = Gq(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 (xG)3 = G tends to 1 as |G | 7→ ∞.

(32)

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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 = Gq(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 (xG)3 = G tends to 1 as |G | 7→ ∞.

Eamonn O’Brien The Ore Conjecture

(33)

### 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 = Gq(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 (xG)3 = G tends to 1 as |G | 7→ ∞.

(34)

logo

### 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 = Gq(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 (xG)3 = G tends to 1 as |G | 7→ ∞.

Eamonn O’Brien The Ore Conjecture

(35)

### 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 Ci be a conjugacy class in G with representative xi. The number of solutions to the equationQt

i =1yi = g with yi ∈ Ci is equal to

|C1| · · · |Ct|

|G |

X

χ∈Irr(G )

χ(x1) · · · χ(xt)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

Eamonn O’Brien The Ore Conjecture

(37)

### 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

(38)

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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

Eamonn O’Brien The Ore Conjecture

(39)

### 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 key step

X

χ∈Irr(G)

|χ(g )|2 = |CG(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.

Eamonn O’Brien The Ore Conjecture

(41)

### The key step

X

χ∈Irr(G)

|χ(g )|2 = |CG(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.

(42)

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### The key step

X

χ∈Irr(G)

|χ(g )|2 = |CG(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.

Eamonn O’Brien The Ore Conjecture

(43)

### 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.

(44)

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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.

Eamonn O’Brien The Ore Conjecture

(45)

### 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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### Earlier work on Thompson / Ore

 Ore (1951): conjectured and proved Ore for An.

 Hs¨u (1965): Thompson for An.

 R.C. Thompson (1962-63): Ore for PSLn(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): PSpn(q), q ≡ 1 mod 4.

Eamonn O’Brien The Ore Conjecture

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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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 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.

Eamonn O’Brien The Ore Conjecture

(49)

 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.

(50)

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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.

Eamonn O’Brien The Ore Conjecture

(51)

### 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

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.

Eamonn O’Brien The Ore Conjecture

(53)

### 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.

(54)

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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.

Eamonn O’Brien The Ore Conjecture

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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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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.

Eamonn O’Brien The Ore Conjecture

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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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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.

Eamonn O’Brien The Ore Conjecture

(59)

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.

Eamonn O’Brien The Ore Conjecture

(61)

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

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.

Eamonn O’Brien The Ore Conjecture

(63)

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

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.

Eamonn O’Brien The Ore Conjecture

(65)

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.

(66)

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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.

Eamonn O’Brien The Ore Conjecture

(67)

### 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 PSUn(7).

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

(68)

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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 PSUn(7).

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

Eamonn O’Brien The Ore Conjecture

(69)

### 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 PSUn(7).

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

(70)

logo

### 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 PSUn(7).

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

Eamonn O’Brien The Ore Conjecture

(71)

### 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 Cln(q) for n ≤ n0.

(72)

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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 Cln(q) for n ≤ n0.

Eamonn O’Brien The Ore Conjecture

(73)

### 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 Cln(q) for n ≤ n0.

(74)

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2n

### (2)

Lemma

Assume n ≥ 7, and let x be an unbreakable element of G = Sp(V ) = Sp2n(2). Then |CG(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.

Eamonn O’Brien The Ore Conjecture

(75)

2n

### (2)

Lemma

Assume n ≥ 7, and let x be an unbreakable element of G = Sp(V ) = Sp2n(2). Then |CG(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.

(76)

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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)(q2(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.

Eamonn O’Brien The Ore Conjecture

(77)

### Some facts

 P

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

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

X

χ∈ Irr(G), χ(1)≥N

|χ(g )|

χ(1) ≤ k(G )1/2|CG(g )|1/2

N .

(78)

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### Some facts

 P

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

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

X

χ∈ Irr(G), χ(1)≥N

|χ(g )|

χ(1) ≤ k(G )1/2|CG(g )|1/2

N .

Eamonn O’Brien The Ore Conjecture

(79)

We can readily bound S2(x ).

Lemma

Suppose n ≥ 7. If |CG(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 · 2n/2|CG(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.

(80)

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We can readily bound S2(x ).

Lemma

Suppose n ≥ 7. If |CG(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 · 2n/2|CG(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.

Eamonn O’Brien The Ore Conjecture

(81)

We can readily bound S2(x ).

Lemma

Suppose n ≥ 7. If |CG(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 · 2n/2|CG(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.

(82)

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We can readily bound S2(x ).

Lemma

Suppose n ≥ 7. If |CG(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 · 2n/2|CG(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.

Eamonn O’Brien The Ore Conjecture

(83)

Lemma

Suppose n ≥ 7. If |CG(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).

(84)

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### The induction base

Some very hard base cases where Ore must be verified directly:

e.g. Sp(12, q), Ω11(3), SU6(7)

In most cases, directly verified the conjecture by constructing character table using Unger algorithm as implemented in Magma. Variations needed for Sp16(2).

For unitary groups: certain equations solved explicitly by finding elements which satisfy these.

About 3 years of CPU time.

Eamonn O’Brien The Ore Conjecture

(85)

### The induction base

Some very hard base cases where Ore must be verified directly:

e.g. Sp(12, q), Ω11(3), SU6(7)

In most cases, directly verified the conjecture by constructing character table using Unger algorithm as implemented in Magma.

Variations needed for Sp16(2).

For unitary groups: certain equations solved explicitly by finding elements which satisfy these.

About 3 years of CPU time.

(86)

logo

### The induction base

Some very hard base cases where Ore must be verified directly:

e.g. Sp(12, q), Ω11(3), SU6(7)

In most cases, directly verified the conjecture by constructing character table using Unger algorithm as implemented in Magma.

Variations needed for Sp16(2).

For unitary groups: certain equations solved explicitly by finding elements which satisfy these.

About 3 years of CPU time.

Eamonn O’Brien The Ore Conjecture

(87)

### The induction base

Some very hard base cases where Ore must be verified directly:

e.g. Sp(12, q), Ω11(3), SU6(7)

In most cases, directly verified the conjecture by constructing character table using Unger algorithm as implemented in Magma.

Variations needed for Sp16(2).

For unitary groups: certain equations solved explicitly by finding elements which satisfy these.

About 3 years of CPU time.

(88)

logo

### The induction base

Some very hard base cases where Ore must be verified directly:

e.g. Sp(12, q), Ω11(3), SU6(7)

In most cases, directly verified the conjecture by constructing character table using Unger algorithm as implemented in Magma.

Variations needed for Sp16(2).

For unitary groups: certain equations solved explicitly by finding elements which satisfy these.

About 3 years of CPU time.

Eamonn O’Brien The Ore Conjecture

(89)

### The infinite context

Every element is a commutator:

Goto (1949): in a connected compact semisimple group.

Pasiencier & Wang (1962): in a semisimple algebraic group over C. Ree (1964): in a connected semisimple algebraic group defined over an algebraically closed field.

(90)

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### The infinite context

Every element is a commutator:

Goto (1949): in a connected compact semisimple group.

Pasiencier & Wang (1962): in a semisimple algebraic group over C. Ree (1964): in a connected semisimple algebraic group defined over an algebraically closed field.

Eamonn O’Brien The Ore Conjecture

(91)

### The infinite context

Every element is a commutator:

Goto (1949): in a connected compact semisimple group.

Pasiencier & Wang (1962): in a semisimple algebraic group over C.

Ree (1964): in a connected semisimple algebraic group defined over an algebraically closed field.

(92)

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### The infinite context

Every element is a commutator:

Goto (1949): in a connected compact semisimple group.

Pasiencier & Wang (1962): in a semisimple algebraic group over C.

Ree (1964): in a connected semisimple algebraic group defined over an algebraically closed field.

Eamonn O’Brien The Ore Conjecture

(93)

### A related question

Problem

Can every element of a finite simple group be obtained as a commutator of a generating pair?

No! Only 44 of the elements of A5 can be obtained in this way; 146 elements of PSL(2, 7).

McCullough & Wanderley: true for PSL(2, q) for q ≥ 11. Garrion & Shalev (2009): “almost every” element is obtained as commutator of a generating pair.

(94)

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### A related question

Problem

Can every element of a finite simple group be obtained as a commutator of a generating pair?

No! Only 44 of the elements of A5 can be obtained in this way;

146 elements of PSL(2, 7).

McCullough & Wanderley: true for PSL(2, q) for q ≥ 11. Garrion & Shalev (2009): “almost every” element is obtained as commutator of a generating pair.

Eamonn O’Brien The Ore Conjecture

(95)

### A related question

Problem

Can every element of a finite simple group be obtained as a commutator of a generating pair?

No! Only 44 of the elements of A5 can be obtained in this way;

146 elements of PSL(2, 7).

McCullough & Wanderley: true for PSL(2, q) for q ≥ 11.

Garrion & Shalev (2009): “almost every” element is obtained as commutator of a generating pair.

(96)

logo

### A related question

Problem

Can every element of a finite simple group be obtained as a commutator of a generating pair?

No! Only 44 of the elements of A5 can be obtained in this way;

146 elements of PSL(2, 7).

McCullough & Wanderley: true for PSL(2, q) for q ≥ 11.

Garrion & Shalev (2009): “almost every” element is obtained as commutator of a generating pair.

Eamonn O’Brien The Ore Conjecture

generalization of Weyl’s gauge invariance, to a possible new theory of interactions

Matrix model recursive formulation of 1/N expansion: all information encoded in spectral curve ⇒ generates topological string amplitudes... This is what we

Define instead the imaginary.. potential, magnetic field, lattice…) Dirac-BdG Hamiltonian:. with small, and matrix

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• involves teaching how to connect the sounds with letters or groups of letters (e.g., the sound /k/ can be represented by c, k, ck or ch spellings) and teaching students to