Frank Lübeck

# Elements of order 2 and 3 in Exceptional Groups of Lie Type

Let p be a prime and G be a simple algebraic group of exceptional type over the algebraic closure of a finite field with p elements. For each Frobenius morphism of G and corresponding power q of p we consider the finite groups of Lie type G(q). Here, we describe the conjugacy classes of elements of order 2 and 3 in the groups G(q).

Notation: Below unipotent classes and centralizers of semisimple elements are parametrized in terms of root systems. We use the convention that a root subsystem of systems with roots of different lengths are preceded by a tilde (like `~A_1`) if the subsystem consists of short roots only.

## Elements of order 2 (resp. 3) in the case p=2 (resp. 3)

In this case elements of order 2 (3) are unipotent. Unipotent classes are parametrized for all types of groups. The element orders can be found from the size of the largest Jordan block in some representation in defining characteristic. These can be read off from

Lawther, R. Correction to: "Jordan block sizes of unipotent elements in exceptional algebraic groups" [Comm. Algebra 23 (1995), no. 11, 4125--4156]. Comm. Algebra 26 (1998), no. 8, 2709.

For p=2 it turns out that a non-trivial class contains elements of order 2 if and only if their Bala-Carter parameter only involves root systems of type A_1 or ~A_1.

Similarly, for p=3, those non-trivial classes have elements of order 3 which have a parameter only involving root systems of type A_1, ~A_1, A_2, ~A_2 or G_2(a_1).

Here is a table of such classes with minimal centralizer dimension.

Type of G       p             class u^G             dim C_G(u)
G_2 2 ~A_1 6
3 G_2(a_1) 4
F_4 2 A_1 + ~A_1 24
3 ~A_2 + A_1 16
E_6 2 3 A_1 38
3 2 A_2 + A_1 24
E_7 2 4 A_1 63
3 2 A_2 + A_1 43
E_8 2 4 A_1 120
3 2 A_2 + 2 A_1 80

All of the classes mentioned in this table have representatives in all corresponding groups G(q). The intersection of G_2(a_1) with G(q) contains two G(q)-conjugacy classes. In all other cases the elements lying in G(q) form a single G(q)-class.

## Semisimple elements of order 2 and 3

For a given root datum and a fixed maximal torus T one can write down the finite number of elements in T of a given fixed order and then determine their centralizers in terms of a root subsystem and the Weyl group of G. Having the conjugacy classes of elements of order 2 and 3 in the algebraic groups one has to determime the possible structures of the centralizers in G(q). I have worked out the details using the software packages GAP and CHEVIE. More details on the computational setup can be found in Section 2 of:

Brunat, O. and Lübeck, F., On defining characteristic representations of finite reductive groups. Journal of Algebra 395 (2013), 121-141.

The results are given type by type. For the centralizer orders we use the Phi<i> notation for the i-th cyclotomic polynomial evaluated at the prime power q, here is an explicit list. Polynomial parts in the type descriptions refer to tori of that order. The dimensions of the centralizers in the algebraic group are given by the polynomial degree of the orders of the centralizers. (The computer programs used here also allow one to compute similar information for other "small" orders.)

Remark: For the conjugacy classes mentioned for groups of adjoint type E6(q)ad, 2E6(q)ad and E7(q)ad we mention when they are not contained in the commutator subgroup (or equivalently, when they are not images of conjugacy classes in the simply connected group of the same type under the simply connected covering).

### 2B2(q2) (Suzuki groups)

Here elements of order 2 are unipotent and there are no elements of order 3.

### G2(q) and 2G2(q2) (Ree groups)

Elements of order 2: There is one class with centralizer of type
A_1(q) + ~A_1(q) and of order q^2 Phi1^2 Phi2^2.
In ^2G_2 the centralizer is of type
A_1(q^2) with order q^2 Phi1 Phi2 Phi4.

Elements of order 3: There is one class with centralizer
A_2(q) of order q^3 Phi1^2 Phi2 Phi3, if q = 1 mod 3 and
^2A_2(q) of order q^3 Phi1 Phi2^2 Phi6, if q = 2 mod 3.

There is one class with centralizer
~A_1(q) + (q-1) of order q Phi1^2 Phi2, if q = 1 mod 3
~A_1(q) + (q+1) of order q Phi1 Phi2^2, if q = 2 mod 3.

### 3D4(q) (Steinberg triality groups)

These groups can be considered as subgroups of the simply-connected or the adjoint algebraic group of type D4, the description here is the same in both cases.

Elements of order 2: There is one class with centralizer
A_1(q^3) + A_1(q) of order q^4*Phi1^2*Phi2^2*Phi3*Phi6.

Elements of order 3: There is one class with centralizer
A_1(q^3) + (q-1) of order q^3*Phi1^2*Phi2*Phi3*Phi6, if q = 1 mod 3 and
A_1(q^3) + (q+1) of order q^3*Phi1*Phi2^2*Phi3*Phi6, if q = 2 mod 3
and one class with centralizer
A_2(q) + (q^2+q+1) of order q^3*Phi1^2*Phi2*Phi3^2, if q = 1 mod 3 and
^2A_2(q) + (q^2-q+1) of order q^3*Phi1*Phi2^2*Phi6^2, if q = 2 mod 3.

### F4(q) and 2F4(q2) (Ree groups)

Elements of order 2: There is one class with centralizer
B_4(q) of order q^16 Phi1^4 Phi2^4 Phi3 Phi4^2 Phi6 Phi8
and one class with centralizer
C_3(q) + A_1(q) of order q^10 Phi1^4 Phi2^4 Phi3 Phi4 Phi6.

Elements of order 3: There is one class with centralizer
A_2(q) + ~A_2(q) of order q^6 Phi1^4 Phi2^2 Phi3^2, if q = 1 mod 3
^2A_2(q) + ^2~A_2(q) of order q^6 Phi1^2 Phi2^4 Phi6^2, if q = 2 mod 3
and one class with centralizer
B_3(q) + (q-1) of order q^9 Phi1^4 Phi2^3 Phi3 Phi4 Phi6, if q = 1 mod 3
B_3(q) + (q+1) of order q^9 Phi1^3 Phi2^4 Phi3 Phi4 Phi6, if q = 2 mod 3
and one class with centralizer
C_3(q) + (q-1) of order q^9 Phi1^4 Phi2^3 Phi3 Phi4 Phi6, if q = 1 mod 3
C_3(q) + (q+1) of order q^9 Phi1^3 Phi2^4 Phi3 Phi4 Phi6, if q = 2 mod 3.
In ^2F_4 only the first of the classes above occurs, in that case the centralizer is of type
^2A_2(q^2) with order q^6 Phi1 Phi2 Phi4^2 Phi12.

Elements of order 2: There is one class with centralizer
A_5(q) + A_1(q) of order q^16 Phi1^6 Phi2^4 Phi3^2 Phi4 Phi5 Phi6
and one class with centralizer
D_5(q) + (q-1) of order q^20 Phi1^6 Phi2^4 Phi3 Phi4^2 Phi5 Phi6 Phi8.

Elements of order 3: There are two classes with centralizer
D_5(q) + (q-1) of order q^20 Phi1^6 Phi2^4 Phi3 Phi4^2 Phi5 Phi6 Phi8, if q = 1 mod 3
(these classes are in the derived subgroup only if q = 1 mod 9)
and two classes with centralizer
A_4(q) + A_1(q) + (q-1) of order q^11 Phi1^6 Phi2^3 Phi3 Phi4 Phi5, if q = 1 mod 3
(these classes are in the derived subgroup only if q = 1 mod 9)
and one class with centralizer
A_5(q) + (q-1) of order q^15 Phi1^6 Phi2^3 Phi3^2 Phi4 Phi5 Phi6, if q = 1 mod 3
A_5(q) + (q+1) of order q^15 Phi1^5 Phi2^4 Phi3^2 Phi4 Phi5 Phi6, if q = 2 mod 3.
Then there is one geometric class with centralizer of type (A_2 + A_2 + A_2).3, splitting as follows:
For q = 1 mod 3: in one G(q)-class with centralizer
(A_2(q) + A_2(q) + A_2(q)).3 of order 3 q^9 Phi1^6 Phi2^3 Phi3^3 and two G(q)-classes with centralizer
A_2(q^3).3 of order 3 q^9 Phi1^2 Phi2 Phi3^2 Phi6 Phi9.
(the last two G(q)-classes are not in the derived group)
For q = 2 mod 3 in one G(q)-class with centralizer
A_2(q^2) + ^2A_2(q) of order q^9 Phi1^3 Phi2^4 Phi3 Phi4 Phi6^2.
And one geometric class with centralizer of type (D_4).3, splitting as follows:
For q = 1 mod 3 in one G(q)-class with centralizer
(D_4(q) + (q^2 - 2*q + 1)).3 of order 3 q^12 Phi1^6 Phi2^4 Phi3 Phi4^2 Phi6 and two G(q)-classes with centralizer
(^3D_4(q) + (q^2 + q + 1)).3 of order 3 q^12 Phi1^2 Phi2^2 Phi3^3 Phi6^2 Phi12.
(the last two G(q)-classes are not in the derived group)
For q = 2 mod 3 in one G(q)-class with centralizer
^2D_4(q) + (q^2-1) of order q^12 Phi1^4 Phi2^4 Phi3 Phi4 Phi6 Phi8.

Elements of order 2: There is one class with centralizer
^2A_5(q) + A_1(q) of order q^16 Phi1^4 Phi2^6 Phi3 Phi4 Phi6^2 Phi10
and one class with centralizer
^2D_5(q) + (q+1) of order q^20 Phi1^4 Phi2^6 Phi3 Phi4^2 Phi6 Phi8 Phi10.

Elements of order 3: There are two classes with centralizer
^2D_5(q) + (q+1) of order q^20 Phi1^4 Phi2^6 Phi3 Phi4^2 Phi6 Phi8 Phi10, if q = 2 mod 3
(these classes are in the derived subgroup only if q = -1 mod 9)
and two classes with centralizer
^2A_4(q) + A_1(q) + (q+1) of order q^11 Phi1^3 Phi2^6 Phi4 Phi6 Phi10, if q = 2 mod 3
(these classes are in the derived subgroup only if q = -1 mod 9)
and one class with centralizer
^2A_5(q) + (q-1) of order q^15 Phi1^4 Phi2^5 Phi3 Phi4 Phi6^2 Phi10, if q = 1 mod 3
^2A_5(q) + (q+1) of order q^15 Phi1^3 Phi2^6 Phi3 Phi4 Phi6^2 Phi10, if q = 2 mod 3.
Then there is one geometric class with centralizer of type (A_2 + A_2 + A_2).3, splitting as follows:
For q = 2 mod 3 in one G(q)-class with centralizer
(^2A_2(q) + ^2A_2(q) + ^2A_2(q)).3 of order 3 q^9 Phi1^3 Phi2^6 Phi6^3
and two G(q)-classes with centralizer
^2A_2(q^3).3 of order 3 q^9 Phi1 Phi2^2 Phi3 Phi6^2 Phi18.
(the last two G(q)-classes are not in the derived group)
For q = 1 mod 3 in one G(q)-class with centralizer
A_2(q^2) + A_2(q) of order q^9 Phi1^4 Phi2^3 Phi3^2 Phi4 Phi6.
And one geometric class with centralizer of type (D_4).3, splitting as follows:
For q = 2 mod 3 in one G(q)-class with centralizer
(D_4(q) + (q^2 + 2*q + 1)).3 of order 3 q^12 Phi1^4 Phi2^6 Phi3 Phi4^2 Phi6
and two G(q)-classes with centralizer
(^3D_4(q) + (q^2 - q + 1)).3 of order 3 q^12 Phi1^2 Phi2^2 Phi3^2 Phi6^3 Phi12.
(the last two G(q)-classes are not in the derived group)
For q = 1 mod 3 in one G(q)-class with centralizer
^2D_4(q) + (q^2 - 1) of order q^12 Phi1^4 Phi2^4 Phi3 Phi4 Phi6 Phi8.

### E6(q)sc (simply connected)

Elements of order 2: Same description as for adjoint groups of type E_6.

Elements of order 3: There are two classes of central elements, centralizer
E_6(q) of order q^36 Phi1^6 Phi2^4 Phi3^3 Phi4^2 Phi5 Phi6^2 Phi8 Phi9 Phi12, if q = 1 mod 3
and one class with centralizer
A_2(q) + A_2(q) + A_2(q) of order q^9 Phi1^6 Phi2^3 Phi3^3, if q = 1 mod 3
A_2(q^2) + ^2A_2(q) of order q^9 Phi1^3 Phi2^4 Phi3 Phi4 Phi6^2, if q = 2 mod 3
and one class with centralizer
D_4(q) + (q^2-2*q+1) of order q^12 Phi1^6 Phi2^4 Phi3 Phi4^2 Phi6, if q = 1 mod 3
^2D_4(q) + (q^2-1) of order q^12 Phi1^4 Phi2^4 Phi3 Phi4 Phi6 Phi8, if q = 2 mod 3.
Then for q = 1 mod 3 there are three classes with centralizer
A_5(q) + (q-1) of order q^15 Phi1^6 Phi2^3 Phi3^2 Phi4 Phi5 Phi6.
And for q = 2 mod 3 there is one class with centralizer
A_5(q) + (q+1) of order q^15 Phi1^5 Phi2^4 Phi3^2 Phi4 Phi5 Phi6.

### 2E6(q)sc (simply connected)

Elements of order 2: Same description as for adjoint groups of type ^2E_6.

Elements of order 3: There are two classes of central elements, centralizer
^2E_6(q) of order q^36 Phi1^4 Phi2^6 Phi3^2 Phi4^2 Phi6^3 Phi8 Phi10 Phi12 Phi18, if q = 2 mod 3
and one class with centralizer
^2A_2(q) + ^2A_2(q) + ^2A_2(q) of order q^9 Phi1^3 Phi2^6 Phi6^3, if q = 2 mod 3
A_2(q^2) + A_2(q) of order q^9 Phi1^4 Phi2^3 Phi3^2 Phi4 Phi6, if q = 1 mod 3
and one class with centralizer
D_4(q) + (q^2+2*q+1) of order q^12 Phi1^4 Phi2^6 Phi3 Phi4^2 Phi6, if q = 2 mod 3
^2D_4(q) + (q^2-1) of order q^12 Phi1^4 Phi2^4 Phi3 Phi4 Phi6 Phi8, if q = 1 mod 3.
Then for q = 2 mod 3 there are three classes with centralizer
^2A_5(q) + (q+1) of order q^15 Phi1^3 Phi2^6 Phi3 Phi4 Phi6^2 Phi10.
And for q = 1 mod 3 there is one class with centralizer
^2A_5(q) + (q-1) of order q^15 Phi1^4 Phi2^5 Phi3 Phi4 Phi6^2 Phi10.

Elements of order 2: There is one class with centralizer
D_6(q) + A_1(q) of order q^31 Phi1^7 Phi2^7 Phi3^2 Phi4^2 Phi5 Phi6^2 Phi8 Phi10
and one geometric class with centralizer of type (A_7).2, splitting into:
one G(q)-class with centralizer
A_7(q).2 of order 2 q^28 Phi1^7 Phi2^4 Phi3^2 Phi4^2 Phi5 Phi6 Phi7 Phi8
(this class is in the derived subgroup only if q = 1 mod 4)
and one G(q)-class with centralizer
^2A_7(q).2 of order 2 q^28 Phi1^4 Phi2^7 Phi3 Phi4^2 Phi6^2 Phi8 Phi10 Phi14,
(this class is in the derived subgroup only if q = -1 mod 4)
and one geometric class with centralizer of type (E_6).2, splitting into:
one G(q)-class with centralizer
(E_6(q) + (q-1)).2 of order 2 q^36 Phi1^7 Phi2^4 Phi3^3 Phi4^2 Phi5 Phi6^2 Phi8 Phi9 Phi12
(this class is in the derived subgroup only if q = 1 mod 4)
and one G(q)-class with centralizer
(^2E_6(q) + (q+1)).2 of order 2 q^36 Phi1^4 Phi2^7 Phi3^2 Phi4^2 Phi6^3 Phi8 Phi10 Phi12 Phi18
(this class is in the derived subgroup only if q = -1 mod 4).

Elements of order 3: There is one class with centralizer
A_5(q) + A_2(q) of order q^18 Phi1^7 Phi2^4 Phi3^3 Phi4 Phi5 Phi6, if q = 1 mod 3
^2A_5(q) + ^2A_2(q) of order q^18 Phi1^4 Phi2^7 Phi3 Phi4 Phi6^3 Phi10, if q = 2 mod 3
and one class with centralizer
E_6(q) + (q-1) of order q^36 Phi1^7 Phi2^4 Phi3^3 Phi4^2 Phi5 Phi6^2 Phi8 Phi9 Phi12, if q = 1 mod 3
^2E_6(q) + (q+1) of order q^36 Phi1^4 Phi2^7 Phi3^2 Phi4^2 Phi6^3 Phi8 Phi10 Phi12 Phi18, if q = 2 mod 3
and one class with centralizer
D_5(q) + A_1(q) + (q-1) of order q^21 Phi1^7 Phi2^5 Phi3 Phi4^2 Phi5 Phi6 Phi8, if q = 1 mod 3
^2D_5(q) + A_1(q) + (q+1) of order q^21 Phi1^5 Phi2^7 Phi3 Phi4^2 Phi6 Phi8 Phi10, if q = 2 mod 3
and one class with centralizer
A_6(q) + (q-1) of order q^21 Phi1^7 Phi2^3 Phi3^2 Phi4 Phi5 Phi6 Phi7, if q = 1 mod 3
^2A_6(q) + (q+1) of order q^21 Phi1^3 Phi2^7 Phi3 Phi4 Phi6^2 Phi10 Phi14, if q = 2 mod 3
and one class with centralizer
D_6(q) + (q-1) of order q^30 Phi1^7 Phi2^6 Phi3^2 Phi4^2 Phi5 Phi6^2 Phi8 Phi10, if q = 1 mod 3
D_6(q) + (q+1) of order q^30 Phi1^6 Phi2^7 Phi3^2 Phi4^2 Phi5 Phi6^2 Phi8 Phi10, if q = 2 mod 3.

### E7(q)sc (simply connected)

Elements of order 2: There is one central class with centralizer
E_7(q) of order q^63 Phi1^7 Phi2^7 Phi3^3 Phi4^2 Phi5 Phi6^3 Phi7 Phi8 Phi9 Phi10 Phi12 Phi14 Phi18
and there are two classes with centralizer
D_6(q) + A_1(q) of order q^31 Phi1^7 Phi2^7 Phi3^2 Phi4^2 Phi5 Phi6^2 Phi8 Phi10.

Elements of order 3: This is the same as for the adjoint groups of type E_7.

### E8(q)

Elements of order 2: There is one class with centralizer
E_7(q) + A_1(q) of order q^64 Phi1^8 Phi2^8 Phi3^3 Phi4^2 Phi5 Phi6^3 Phi7 Phi8 Phi9 Phi10 Phi12 Phi14 Phi18
and one class with centralizer
D_8(q) of order q^56 Phi1^8 Phi2^8 Phi3^2 Phi4^4 Phi5 Phi6^2 Phi7 Phi8^2 Phi10 Phi12 Phi14.

Elements of order 3: There is one class with centralizer
E_6(q) + A_2(q) of order q^39 Phi1^8 Phi2^5 Phi3^4 Phi4^2 Phi5 Phi6^2 Phi8 Phi9 Phi12, if q = 1 mod 3
^2E_6(q) + ^2A_2(q) of order q^39 Phi1^5 Phi2^8 Phi3^2 Phi4^2 Phi6^4 Phi8 Phi10 Phi12 Phi18, if q = 2 mod 3
and one class with centralizer
A_8(q) of order q^36 Phi1^8 Phi2^4 Phi3^3 Phi4^2 Phi5 Phi6 Phi7 Phi8 Phi9, if q = 1 mod 3
^2A_8(q) of order q^36 Phi1^4 Phi2^8 Phi3 Phi4^2 Phi6^3 Phi8 Phi10 Phi14 Phi18, if q = 2 mod 3
and one class with centralizer
E_7(q) + (q-1) of order q^63 Phi1^8 Phi2^7 Phi3^3 Phi4^2 Phi5 Phi6^3 Phi7 Phi8 Phi9 Phi10 Phi12 Phi14 Phi18, if q = 1 mod 3
E_7(q) + (q+1) of order q^63 Phi1^7 Phi2^8 Phi3^3 Phi4^2 Phi5 Phi6^3 Phi7 Phi8 Phi9 Phi10 Phi12 Phi14 Phi18, if q = 2 mod 3
and one class with centralizer
D_7(q) + (q-1) of order q^42 Phi1^8 Phi2^6 Phi3^2 Phi4^3 Phi5 Phi6^2 Phi7 Phi8 Phi10 Phi12, if q = 1 mod 3
^2D_7(q) + (q+1) of order q^42 Phi1^6 Phi2^8 Phi3^2 Phi4^3 Phi5 Phi6^2 Phi8 Phi10 Phi12 Phi14, if q = 2 mod 3.

Last updated: Mon Jul 12 23:29:18 2021 (CET)