Orders of inner-diagonal automorphisms of some simple groups of Lie type

Let 𝐺 be the group of inner-diagonal automorphisms of the simple group E 6 ε ⁢ ( q ) , where ε ∈ { + , - } , 𝑞 is a power of a prime 𝑝, and 3 divides q - ε . Define the set ν ⁢ ( G ) to be the union of the following sets:

1. { q 6 - 1 3 , q 6 + ε ⁢ q 3 + 1 , ( q 2 + ε ⁢ q + 1 ) ⁢ ( q 4 - q 2 + 1 ) , ( q 2 - 1 ) ⁢ ( q 4 + 1 ) , ( q + ε ) ⁢ ( q 5 - ε ) , ( q - ε ) ⁢ ( q 2 + 1 ) ⁢ ( q 3 + ε ) } ;

2. p ⋅ { q 6 - 1 q - ε , q 5 - ε , ( q 3 + ε ) ⁢ ( q - ε ) } ;

3. p ⁢ ( 2 ) ⋅ { ( q 3 - ε ) ⁢ ( q + ε ) , q 4 + q 2 + 1 , q 4 - 1 } ;

4. p ⁢ ( 5 ) ⋅ { q 2 - 1 , q 2 + ε ⁢ q + 1 } ;

5. p ⁢ ( 7 ) ⋅ { q - ε } ;

6. { p ⁢ ( 11 ) } .

Let SL n ε ⁢ ( q ) ⩽ H ⩽ GL n ε ⁢ ( q ) , where n ⩾ 2 , ε ∈ { + , - } , and 𝑞 is a power of a prime 𝑝. Suppose that the index of 𝐻 in GL n ⁢ ( q ) is equal to d 1 and 𝑍 is a subgroup of the center of 𝐻 of order d 2 . Then d 2 divides ( n ⁢ ( q - ε ) d 1 , q - ε ) = | Z ⁢ ( H ) | and ω ⁢ ( H / Z ) consists of the divisors of the following numbers:

1. q n - ε n d 1 ⁢ d 2 ;

2. [ q n 1 - ε n 1 , q n 2 - ε n 2 ] ( d 1 , d 2 , n / ( n 1 , n 2 ) ) for n 1 , n 2 > 0 such that n 1 + n 2 = n ;

3. [ q n 1 - ε n 1 , q n 2 - ε n 2 , … , q n s - ε n s ] for s ⩾ 3 and n 1 , n 2 , … , n s > 0 such that n 1 + n 2 + ⋯ + n s = n ;

4. p k ⁢ q n 1 - ε n 1 ( d 1 , n 1 ⁢ d 2 , n ) for k , n 1 > 0 such that p k - 1 + 1 + n 1 = n ;

5. p k ⁢ [ q n 1 - ε n 1 , q n 2 - ε n 2 , … , q n s - ε n s ] for s ⩾ 2 and k , n 1 , n 2 ⁢ … , n s > 0 such that p k - 1 + 1 + n 1 + n 2 + ⋯ + n s = n ;

6. p k ⁢ q - ε d 1 ⁢ d 2 ⁢ ( n , d 1 ) if p k - 1 + 1 = n for k > 0 ;

7. p ⁢ q 2 - 1 d 1 ⁢ d 2 ⁢ ( 2 , d 1 ) if n = 4 .

Let PSL n ε ⁢ ( q ) ⩽ G ⩽ PGL n ε ⁢ ( q ) , where n ⩾ 2 , ε ∈ { + , - } , and 𝑞 is a power of a prime 𝑝. If the index of 𝐺 in PGL n ε ⁢ ( q ) is equal to 𝑑, then ω ⁢ ( G ) consists of the divisors of the following numbers:

1. q n - ε n d ⁢ ( q - ε ) ;

2. [ q n 1 - ε n 1 , q n 2 - ε n 2 ] ( d , n / ( n 1 , n 2 ) ) for n 1 , n 2 > 0 such that n 1 + n 2 = n ;

3. [ q n 1 - ε n 1 , q n 2 - ε n 2 , … , q n s - ε n s ] for s ⩾ 3 and n 1 , n 2 , … , n s > 0 such that n 1 + n 2 + ⋯ + n s = n ;

4. p k ⁢ q n 1 - ε n 1 d for k , n 1 > 0 such that p k - 1 + 1 + n 1 = n ;

5. p k ⁢ [ q n 1 - ε n 1 , q n 2 - ε n 2 , … , q n s - ε n s ] for s ⩾ 2 and k , n 1 , n 2 ⁢ … , n s > 0 such that p k - 1 + 1 + n 1 + n 2 + ⋯ + n s = n ;

6. p k if p k - 1 + 1 = n for k > 0 .

Let Z 1 be the Hall subgroup of Z ⁢ ( SL n ⁢ ( F ¯ ) ) of order 𝑚, and let 𝜎 be a Frobenius endomorphism of SL n ⁢ ( F ¯ ) such that ( SL n ⁢ ( F ¯ ) ) σ = SL n ε ⁢ ( q ) . Then the spectrum of ( SL n ⁢ ( F ¯ ) / Z 1 ) σ consists of the divisors of the following numbers:

1. q n - ε n q - ε ;

2. [ q n 1 - ε n 1 , q n 2 - ε n 2 , … , q n s - ε n s ] for s ⩾ 2 and n 1 , n 2 , … , n s > 0 such that n 1 + n 2 + ⋯ + n s = n ;

3. p k ⁢ q n 1 - ε n 1 ( n , q - ε , n 1 ) m ′ for k , n 1 > 0 such that p k - 1 + 1 + n 1 = n ;

4. p k ⁢ [ q n 1 - ε n 1 , q n 2 - ε n 2 , … , q n s - ε n s ] for s ⩾ 2 and k , n 1 , n 2 ⁢ … , n s > 0 such that p k - 1 + 1 + n 1 + n 2 + ⋯ + n s = n ;

5. ( n , q - ε ) m ′ ⁢ p k if p k - 1 + 1 = n for k > 0 ;

6. 2 ⁢ p ⁢ ( q + ε ) if n = 4 and m = 1 .

Let a ⩾ 2 , k ⩾ 1 be positive integers and ε ∈ { + , - } .

1. If 𝑝 is an odd prime divisor of a - ε , then ( a k - ε k ) r = ( k ) r ⁢ ( a - ε ) r .

2. If a ≡ ε ( mod 4 ) or 𝑘 is odd, then ( a k - ε k ) 2 = ( k ) 2 ⁢ ( a - ε ) 2 .

3. If a ≡ - ε ( mod 4 ) and 𝑘 is even, then ( a k - ε k ) 2 = ( k ) 2 ⁢ ( a + ε ) 2 .

If 𝑎, 𝑏, and 𝑐 are positive integers, 𝑏 and 𝑐 divide 𝑎, then

The G ¯ σ -orbits of the set of 𝜎-stable conjugates of R ¯ in G ¯ are in bijective correspondence with 𝜎-conjugacy classes in N W ⁢ ( W 1 ) / W 1 , with bijection inducing by R ¯ g ↦ W 1 ⁢ π ⁢ ( g σ ⁢ g - 1 ) .

Proof

See [7, Proposition 3].∎

The group W 1 × W 2 is a normal subgroup of N W ⁢ ( W 1 ) , and W 2 is a normal subgroup of N W ⁢ ( W 1 ) . Also we have the following isomorphisms: N W ⁢ ( W 1 ) / W 1 ≃ N W ⁢ ( Π 1 ) and N W ⁢ ( Π 1 ) / W 2 ≃ A ⁢ u ⁢ t W ⁢ ( Δ 1 ) .

Proof

See [7, Section 2.3]. ∎

Suppose that R ¯ g is 𝜎-stable. Then exp p ⁡ ( ( R ¯ g ) σ ) is equal to the minimal power of 𝑝 greater than the maximal height of a root in Φ 1 .

Proof

This is an easy consequence of [17, Proposition 0.5]. ∎

Suppose that R ¯ g is 𝜎-stable, g ∈ N G ¯ ⁢ ( T ¯ ) , and π ⁢ ( g σ ⁢ g - 1 ) = w . Then Z ⁢ ( ( R ¯ g ) σ ) is conjugate in G ¯ to Z ⁢ ( R ¯ ) ∩ T ¯ σ ∘ w .

Proof

It is not difficult to see that n = g σ ⁢ g - 1 normalizes R ¯ and that we have ( R ¯ g ) σ = R ¯ σ ∘ n . By [8, Proposition 3.6.8], Z ⁢ ( R ¯ σ ∘ n ) is equal to ( Z ⁢ ( R ¯ ) ) σ ∘ w , which is in turn equal to Z ⁢ ( R ¯ ) ∩ T ¯ σ ∘ w . ∎

Let 𝐿 be a finite simple group of Lie type in characteristic 𝑝. Let L ^ = C l ⁢ ( q ) if L = B l ⁢ ( q ) , L ^ = B l ⁢ ( q ) if L = C l ⁢ ( q ) , and L ^ = L otherwise. If L ^ u is the universal version of L ^ , then ω p ′ ⁢ ( Inndiag ⁡ L ) = ω p ′ ⁢ ( L ^ u ) .

Proof

Since Inndiag ⁡ L and L ^ u are in duality (in the sense of [8, Proposition 4.3.1]), these groups have isomorphic maximal tori by [8, Proposition 4.4.1]. ∎

Let 𝑇 be a maximal torus of GL n ε ⁢ ( q ) corresponding to a partition n = n 1 + ⋯ + n s , and let 𝑆 be the image of T ∩ H in 𝐺. Then exp ⁡ ( S ) is equal to

1. exp ⁡ ( T ) / ( d 1 ⁢ d 2 ) if s = 1 ;

2. exp ⁡ ( T ) / ( d 1 , d 2 , n ( n 1 , n 2 ) ) if s = 2 ;

3. exp ⁡ ( T ) if s > 2 .

Proof

If s = 1 , then 𝑇 is cyclic, so (1) is straightforward. If s > 2 , then by [5, Theorems 2.1 and 2.2], the exponents of 𝑇 and the image of T ∩ SL n ε ⁢ ( q ) in PSL n ε ⁢ ( q ) coincide. This proves (3).

Let s = 2 . For i = 1 , 2 , take the matrix D i to be the diagonal matrix of size n i × n i with elements λ i , λ i ( ε ⁢ q ) , … , λ i ( ε ⁢ q ) n i - 1 along the diagonal, where λ i is an element of F ¯ × of order q n i - ε n i . Then det ⁡ D i = λ i ( ( ε ⁢ q ) n i - 1 ) / ( ε ⁢ q - 1 ) , and hence det ⁡ D i is an element of F ¯ × of order q - ε . We can choose λ 1 and λ 2 so that det ⁡ D 1 and det ⁡ D 2 are equal to the same element 𝜆 of F ¯ × of order q - ε , that is,

Put u = diag ⁡ ( D 1 , I n 2 ) and v = diag ⁡ ( I n 1 , D 2 ) . Then 𝑇 is conjugate in GL n ⁢ ( F ¯ ) to the group ⟨ u ⟩ × ⟨ v ⟩ .

Since det ⁡ u = det ⁡ v = λ , it follows that ( det ⁡ u x ⁢ v y ) ( q - ε ) / d 1 = 1 if and only if d 1 divides x + y , so T ∩ H is conjugate to the group generated by u ⁢ v - 1 and v d 1 . Denote the image of an element g ∈ T ∩ H in 𝑆 by g ¯ . Then exp ⁡ S = [ | u ⁢ v - 1 ¯ | , | v d 1 ¯ | ] . It is easily seen that | v d 1 ¯ | = | v d 1 | = | v | / d 1 .

To calculate | u ⁢ v - 1 ¯ | , observe that ( u ⁢ v - 1 ) x ∈ Z if and only if λ 1 x ⁢ d 2 = λ 2 x ⁢ d 2 = 1 and λ 1 x = λ 2 - x . The first condition implies that 𝑥 is divisible by ( q n 1 - ε n 1 ) / d 2 and ( q n 2 - ε n 2 ) / d 2 . Since ( q n 1 - ε n 1 , q n 2 - ε n 2 ) = q ( n 1 , n 2 ) - ε ( n 1 , n 2 ) , we have that

for some integer 𝑦. Then the second condition is equivalent to ( λ ε ⁢ q - 1 d 2 ) c ⁢ y = 1 , where

Since

and n 1 + n 2 = n , we see that | u ⁢ v - 1 ¯ | = | u ⁢ v - 1 | / ( d 2 , n ( n 1 , n 2 ) ) .

It remains to show that the least common multiple of the numbers a = | v | / d 1 and b = | u ⁢ v - 1 | / ( d 2 , n ( n 1 , n 2 ) ) is equal to

Let r ∈ π ⁢ ( | u ⁢ v - 1 | ) . If 𝑟 is coprime to q - ε or n / ( n 1 , n 2 ) , then ( a ) r divides ( b ) r , and so ( [ a , b ] ) r = ( b ) r = ( f ) r , as required. If 𝑟 divides ( q - ε , n ( n 1 , n 2 ) ) , then ( n 1 ) r = ( n 2 ) r and | v | r = | u | r = | u ⁢ v - 1 | r by Lemma 2.1, so ( [ a , b ] ) r = ( f ) r by Lemma 2.2. ∎

Let 𝑅 be a reductive group of GL n ε ⁢ ( q ) corresponding to a partition n = m 1 ⋅ 1 + n 1 + ⋯ + n s , and let 𝑆 be the image of Z ⁢ ( R ) ∩ H in 𝐺. Then exp ⁡ ( S ) is equal to

1. ( q n 1 - ε n 1 ) / ( d 1 , n 1 ⁢ d 2 , n ) if s = 1 ;

2. [ q n 1 - ε n 1 , … , q n s - ε n s ] if s > 1 .

Proof

Let D = λ ⁢ I m 1 with λ ε ⁢ q - 1 = 1 . For i = 1 , … , s , let D i be the diagonal matrix of size n i × n i with elements λ i , λ i ε ⁢ q , … , λ i ( ε ⁢ q ) n i - 1 along the diagonal, where λ i ( ε ⁢ q ) n i - 1 = 1 . The center of 𝑅 is conjugate in GL n ⁢ ( F ¯ ) to the group of diagonal matrices with blocks D , D 1 , … , D s along the diagonal. Let 𝑇 be the subgroup of Z ⁢ ( R ) conjugated to the subgroup consisting of matrices with D = I m 1 . If s > 1 , then T ∩ H has the same exponent as 𝑇 and intersects 𝑍 trivially. Hence exp ⁡ ( S ) = exp ⁡ ( T ) , as required.

Suppose that s = 1 . Let 𝑢 be a generator of the center of GL n ε ⁢ ( q ) , and let 𝑣 be a generator of 𝑇. Without loss of generality, we may assume that u = det ⁡ v ⁢ I n . Then the intersection Z ⁢ ( R ) ∩ H is conjugate to the group generated by u ⁢ v - n and v d 1 . The order of v d 1 modulo 𝑍 is equal to | v | / d 1 . Since 𝑍 is generated by u ( q - ε ) / d 2 , the order of u ⁢ v - n modulo 𝑍 is equal to [ ( q - ε ) / d 2 , | v | / ( n , | v | ) ] . Thus

By Lemma 2.2, we have

Expressing ( q - ε ) / d 2 as the ratio of | v | and | v | ⁢ d 2 / ( q - ε ) and applying Lemma 2.2 once again, we see that exp ⁡ ( S ) = | v | / ( d 1 , n , | v | ⁢ d 2 / ( q - ε ) ) . Since

it follows that exp ⁡ ( S ) = | v | / ( d 1 , n , n 1 ⁢ d 2 ) , and this proves (1). ∎

Proof of Theorem 2

It is sufficient to prove that, for every reductive subgroup 𝑅 of GL n ε ⁢ ( q ) of maximal rank, the number η ⁢ ( R ) divides some number in items (1)–(7), and every number in items (1)–(7) is equal to η ⁢ ( R ) for some 𝑅.

If 𝑅 is a maximal torus, then η ⁢ ( R ) = exp ⁡ ( ( R ∩ H ) / Z ) is given in Lemma 3.1. These numbers are listed in items (1)–(3).

Suppose that 𝑅 is not a maximal torus. By repeating the argument used in [2, Propositions 2.3 and 2.4], one can show that if n ≠ 4 , then η ⁢ ( R ) divides η ⁢ ( R 1 ) for some reductive group R 1 of maximal rank corresponding to a partition of the form m 1 ⋅ 1 + n 1 + ⋯ + n s . The same is true if n = 4 and 𝑅 is not isomorphic to GL 2 ⁢ ( q 2 ) .

Let 𝑅 be a group corresponding to the partition n = m 1 ⋅ 1 + n 1 + ⋯ + n s . If s ⩾ 1 , then exp ⁡ ( ( Z ⁢ ( R ) ∩ H ) / Z ) is calculated in Lemma 3.2. If s = 0 , then R = GL n ε ⁢ ( q ) , and so