The (2, 3)-generation of the finite 8-dimensional orthogonal groups
-
and
Abstract
We construct
1 Introduction
A group is said to be
By a famous result of Liebeck and Shalev [8], the finite simple classical groups are
According to [11], the following finite simple classical groups are not
Here, given a finite 8-dimensional orthogonal group, we construct plenty of
The groups
2 Notation and preliminary results
Denote by 𝔽 the algebraic closure of the field
Given
The lists of maximal subgroups of
In the following lemma, 𝐻 may be any subgroup of
Let 𝑈 be an 𝐻-invariant subspace of
Proof
Let
Assume further that 𝐻 preserves a non-singular form 𝐽, namely that
It follows that
Given an eigenvalue 𝜆 of a matrix
3 Generators for
Ω
8
+
(
q
)
and the triality automorphism
Since
Then 𝑥 and 𝑦 have respective orders 2 and 3.
Moreover, they belong to the group
The group
Then 𝜏 is defined by the following permutation action [3, Proposition 12.2.3]:
We need 𝜏 mainly because it moves certain maximal subgroups from class
In order to calculate
for
Now set
and then
Finally, we define
Now
and
Applying (3.1), we get
Furthermore,
if
if
If
Proof
We begin by proving the absolute irreducibility of 𝐻.
The eigenspace
Let
has determinant
Let 𝑁 be the matrix whose columns are the 𝑣,
So assume
If
If
In both cases, take the matrix 𝐵 whose columns are 𝑣,
Now, to prove the absolute irreducibility of the groups
Consider
Then we have
where
If
Finally, consider
Then we have
If
Since
Let 𝑝 be odd.
If
Proof
The characteristic polynomial of
By direct computations, if
Note that
Let 𝑈 and
Now suppose that
and the vectors
The matrix
If
Finally, since 1 is an eigenvalue of 𝜂 having algebraic multiplicity at least 4, but geometric multiplicity equal to 2, we conclude that the order of 𝜂 is divisible by 𝑝. ∎
Let 𝑝 be odd.
If
Proof
We write 𝑥 for
Let
and
Then 𝑀 has determinant
Hence, there exists
Let 𝐵 be the matrix whose columns are 𝑢,
Case (1). If
is nonzero unless
It follows that
Case (2). We may assume
where
Let 𝑝 be odd.
If
Proof
We write 𝑥 for
By direct computations, the eigenspace
Let 𝑈 and
where
If
Now suppose
has determinant
If
Proof
Suppose that 𝐻 is contained in one of the maximal subgroups described in the statement.
By [1, Table 8.50] and [6],
Applying the same argument, we can also prove that 𝐻 is neither contained in any maximal subgroup
nor in any maximal subgroup
However, the first type of subgroup will be excluded by Lemma 4.1, while the second type has been already excluded by the absolute irreducibility of 𝐻.
4 The (2, 3)-generation of
Ω
8
+
(
q
)
Suppose
Proof
The characteristic polynomial of
Suppose that our claim is false. Note that 𝐻 is absolutely irreducible by Lemmas 3.1 and 3.2.
Case 1.
Then
Case 2.
Let
and, considering its rational form, that
Setting
Case 3.
Clearly, we must have
The group 𝐻 is not contained in any maximal subgroup of class
Proof
Suppose the contrary.
Then there exists
The group 𝐻 is not contained in any maximal subgroup of type
Proof
Suppose the contrary, and write
First, assume
It follows that
A similar argument can be applied for 𝑝 odd. In this case, we have
When
The subgroup
acts by conjugation on
Each
Proof
Consider first a preimage 𝑔 of
Next, consider a preimage 𝑔 of
The characteristic polynomial of
Suppose
Proof
Clearly,
and for
The group 𝐻 is not contained in any maximal subgroup of type
Proof
Let
Then there exists a non-singular matrix 𝑃 and a scalar
where
The characteristic polynomials of
Suppose
Proof
By the considerations of Section 3, Lemma 3.1 and Lemma 3.2, 𝐻 is an absolutely irreducible subgroup of
5 Generators for
Ω
8
-
(
q
)
5.1 Fields of characteristic 2
We first deal with the case
The group
Proof
Take
Then 𝑥 is an involution, 𝑦 has order 3, and
Proceeding as we will do for the general case, it is easy to prove that
Assume
Then 𝑥 is an involution, 𝑦 has order 3, and
whose associated symmetric form has Gram matrix
Clearly,
For all
Proof
By Carlitz’s formula [2], there exist
The characteristic polynomial of
Note that
The group 𝐻 is absolutely irreducible.
Proof
Let 𝑈 and
If the restriction
Since
Then
5.2 Fields of odd characteristic
Suppose 𝑞 odd.
Let
Then 𝑥 is an involution, 𝑦 has order 3, and
of determinant 𝜉.
By [1, Proposition 1.5.42], the subgroup
The characteristic polynomial of
Note that
The group 𝐻 is absolutely irreducible.
Proof
By direct computations,
Let 𝑈 and
If
Writing
has determinant
So, for some
We will show that
Suppose first
If
which is nonzero unless either (i)
In case (ii), we have
which is nonzero unless
Suppose that
Proof
Suppose the contrary.
Then there exists
it follows that
6 The (2, 3)-generation of
Ω
8
-
(
q
)
We recall that the maximal subgroups of
The group 𝐻 is not contained in any maximal subgroup of class 𝒮.
Proof
The class 𝒮 is non-empty only when
Let
Proof
By the considerations of Section 5.1 and Lemma 5.3, 𝐻 is an absolutely irreducible subgroup of
For the second part of the statement, if
Let
Proof
Take
-
Communicated by: Timothy C. Burness
References
[1] J. N. Bray, D. F. Holt and C. M. Roney-Dougal, The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, London Math. Soc. Lecture Note Ser. 407, Cambridge University, Cambridge, 2013. 10.1017/CBO9781139192576Search in Google Scholar
[2] L. Carlitz, A theorem of Dickson on irreducible polynomials, Proc. Amer. Math. Soc. 3 (1952), 693–700. 10.1090/S0002-9939-1952-0049940-6Search in Google Scholar
[3] R. W. Carter, Simple Groups of Lie Type, John Wiley & Sons, New York, 1972. Search in Google Scholar
[4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups, Oxford University, Eynsham, 1985. Search in Google Scholar
[5] N. Jacobson, Basic Algebra. I, 2nd ed., W.H. Freeman, New York, 1985. Search in Google Scholar
[6]
P. B. Kleidman,
The maximal subgroups of the finite 8-dimensional orthogonal groups
[7] P. B. Kleidman and M. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Note Ser. 129, Cambridge University, Cambridge, 1990. 10.1017/CBO9780511629235Search in Google Scholar
[8]
M. W. Liebeck and A. Shalev,
Classical groups, probabilistic methods, and the
[9]
M. A. Pellegrini,
The
[10]
M A. Pellegrini,
The
[11]
M. A. Pellegrini and M. C. Tamburini,
Finite simple groups of low rank: Hurwitz generation and
[12]
M. A. Pellegrini and M. C. Tamburini Bellani,
The
[13]
M. A. Pellegrini and M. C. Tamburini Bellani,
On the
[14]
M. A. Vsemirnov,
On
[15] R. A. Wilson, The Finite Simple Groups, Grad. Texts in Math. 251, Springer, London, 2009. 10.1007/978-1-84800-988-2Search in Google Scholar
© 2022 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Frontmatter
- Fusion systems with J-components over 𝐹2𝑒 with 𝑒 > 1
- The (2, 3)-generation of the finite 8-dimensional orthogonal groups
- Finite groups admitting a coprime automorphism satisfying an additional polynomial identity
- Conjugacy classes of maximal cyclic subgroups of metacyclic 𝑝-groups
- Free representations of outer automorphism groups of free products via characteristic abelian coverings
Articles in the same Issue
- Frontmatter
- Fusion systems with J-components over 𝐹2𝑒 with 𝑒 > 1
- The (2, 3)-generation of the finite 8-dimensional orthogonal groups
- Finite groups admitting a coprime automorphism satisfying an additional polynomial identity
- Conjugacy classes of maximal cyclic subgroups of metacyclic 𝑝-groups
- Free representations of outer automorphism groups of free products via characteristic abelian coverings
