Affiliation:
1. Sobolev Institute of Mathematics , 4, Koptyug av., 630090 , Novosibirsk , Russia
Abstract
Abstract
We prove that if
L
=
F
4
2
(
2
2
n
+
1
)
′
L={}^{2}F_{4}(2^{2n+1})^{\prime}
and 𝑥 is a nonidentity automorphism of 𝐿, then
G
=
⟨
L
,
x
⟩
G=\langle L,x\rangle
has four elements conjugate to 𝑥 that generate 𝐺.
This result is used to study the following conjecture about the 𝜋-radical of a finite group.
Let 𝜋 be a proper subset of the set of all primes and let 𝑟 be the least prime not belonging to 𝜋.
Set
m
=
r
m=r
if
r
=
2
r=2
or 3 and
m
=
r
−
1
m=r-1
if
r
⩾
5
r\geqslant 5
.
Supposedly, an element 𝑥 of a finite group 𝐺 is contained in the 𝜋-radical
O
π
(
G
)
\operatorname{O}_{\pi}(G)
if and only if every 𝑚 conjugates of 𝑥 generate a 𝜋-subgroup.
Based on the results of this and previous papers, the conjecture is confirmed for all finite groups whose every nonabelian composition factor is isomorphic to a sporadic, alternating, linear, unitary simple group, or to one of the groups of type
B
2
2
(
2
2
n
+
1
)
{}^{2}B_{2}(2^{2n+1})
,
G
2
2
(
3
2
n
+
1
)
{}^{2}G_{2}(3^{2n+1})
,
F
4
2
(
2
2
n
+
1
)
′
{}^{2}F_{4}(2^{2n+1})^{\prime}
,
G
2
(
q
)
G_{2}(q)
, or
D
4
3
(
q
)
{}^{3}D_{4}(q)
.
Funder
Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences
Subject
Algebra and Number Theory
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