Affiliation:
1. Department of Mathematics , London School of Economics and Political Science , Houghton Street , London , WC2A 2AE , United Kingdom
Abstract
Abstract
A Hurwitz generating triple for a group 𝐺 is an ordered triple of elements
(
x
,
y
,
z
)
∈
G
3
(x,y,z)\in G^{3}
, where
x
2
=
y
3
=
z
7
=
x
y
z
=
1
x^{2}=y^{3}=z^{7}=xyz=1
and
⟨
x
,
y
,
z
⟩
=
G
\langle x,y,z\rangle=G
.
For the finite quasisimple exceptional groups of types
F
4
F_{4}
,
E
6
E_{6}
,
E
6
2
{}^{2}E_{6}
,
E
7
E_{7}
and
E
8
E_{8}
, we provide restrictions on which conjugacy classes 𝑥, 𝑦 and 𝑧 can belong to if
(
x
,
y
,
z
)
(x,y,z)
is a Hurwitz generating triple.
We prove that there exist Hurwitz generating triples for
F
4
(
3
)
F_{4}(3)
,
F
4
(
5
)
F_{4}(5)
,
F
4
(
7
)
F_{4}(7)
,
F
4
(
8
)
F_{4}(8)
,
E
6
(
3
)
E_{6}(3)
and
E
7
(
2
)
E_{7}(2)
, and that there are no such triples for
F
4
(
2
3
n
-
2
)
F_{4}(2^{3n-2})
,
F
4
(
2
3
n
-
1
)
F_{4}(2^{3n-1})
,
E
6
(
7
3
n
-
2
)
E_{6}(7^{3n-2})
,
E
6
(
7
3
n
-
1
)
E_{6}(7^{3n-1})
,
S
E
6
(
7
n
)
SE_{6}(7^{n})
or
E
6
2
(
7
n
)
{}^{2}E_{6}(7^{n})
when
n
≥
1
n\geq 1
.
Subject
Algebra and Number Theory
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