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On generations by conjugate elements in almost simple groups with socle
\mbox{}^2F_4(q^2)'
Authors
Danila O. Revin
Andrei V. Zavarnitsine
Publication date
28 December 2022
Publisher
View
on
arXiv
Abstract
We prove that if
L=\mbox{}^2F_4(2^{2n+1})'
and
x
x
x
is a nonidentity automorphism of
L
L
L
then
G
=
β¨
L
,
x
β©
G=\langle L,x\rangle
G
=
β¨
L
,
x
β©
has four elements conjugate to
x
x
x
that generate
G
G
G
. This result is used to study the following conjecture about the
Ο
\pi
Ο
-radical of a finite group: Let
Ο
\pi
Ο
be a proper subset of the set of all primes and let
r
r
r
be the least prime not belonging to
Ο
\pi
Ο
. Set
m
=
r
m=r
m
=
r
if
r
=
2
r=2
r
=
2
or
3
3
3
and set
m
=
r
β
1
m=r-1
m
=
r
β
1
if
r
β©Ύ
5
r\geqslant 5
r
β©Ύ
5
. Supposedly, an element
x
x
x
of a finite group
G
G
G
is contained in the
Ο
\pi
Ο
-radical
O
β‘
Ο
(
G
)
\operatorname{O}_\pi(G)
O
Ο
β
(
G
)
if and only if every
m
m
m
conjugates of
x
x
x
generate a
Ο
\pi
Ο
-subgroup. Based on the results of this paper and a few previous ones, the conjecture is confirmed for all finite groups whose every nonabelian composition factor is isomorphic to a sporadic, alternating, linear, or unitary simple group, or to one of the groups of type
2
B
2
(
2
2
n
+
1
)
{}^2B_2(2^{2n+1})
2
B
2
β
(
2
2
n
+
1
)
,
2
G
2
(
3
2
n
+
1
)
{}^2G_2(3^{2n+1})
2
G
2
β
(
3
2
n
+
1
)
,
2
F
4
(
2
2
n
+
1
)
β²
{}^2F_4(2^{2n+1})'
2
F
4
β
(
2
2
n
+
1
)
β²
,
G
2
(
q
)
G_2(q)
G
2
β
(
q
)
, or
3
D
4
(
q
)
{}^3D_4(q)
3
D
4
β
(
q
)
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oai:arXiv.org:2212.13785
Last time updated on 16/01/2023