Hurwitz Generation in Groups of Types $F_4$, $E_6$, $^2E_6$, $E_7$ and $E_8$

Abstract: A Hurwitz generating triple for a group

G

is an ordered triple of elements

(x, y, z) i n G^{3}

where

x^{2} = y^{3} = z^{7} = x y z = 1

and

l a n g l e x, y, z a n g l e = G

. For the finite quasisimple exceptional groups of types

F_{4}

,

E_{6}

,

2 E_{6}

,

E_{7}

and

E_{8}

, we provide restrictions on which conjugacy classes

x

,

y

and

z

can belong to if

(x, y, z)

is a Hurwitz generating triple. We prove that there exist Hurwitz generating triples for

F_{4} (3)

,

F_{4} (5)

,

F_{4} (7)

,

F_{4} (8)

,

E_{6} (3)

and

E_{7} (2)

, and that there are no such triples for

F_{4} (2^{3 n - 2})

,

F_{4} (2^{3 n - 1})

,

E_{6} (7^{3 n - 2})

,

E_{6} (7^{3 n - 1})

,

S E_{6} (7^{n})

or

2 E_{6} (7^{n})

when

n g e q 1

.

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