Let
G
G
be a group generated by a set
C
C
of involutions which is closed under conjugation. Let
π
\pi
be a set of odd primes. Assume that either (1)
G
G
is solvable, or (2)
G
G
is a linear group.
We show that if the product of any two involutions in
C
C
is a
π
\pi
-element, then
G
G
is solvable in both cases and
G
=
O
π
(
G
)
⟨
t
⟩
G=O_{\pi }(G)\langle t\rangle
, where
t
∈
C
t\in C
.
If (2) holds and the product of any two involutions in
C
C
is a unipotent element, then
G
G
is solvable.
Finally we deduce that if
G
\mathcal {G}
is a sharply
2
2
-transitive (infinite) group of odd (permutational) characteristic, such that every
3
3
involutions in
G
\mathcal {G}
generate a solvable or a linear group; or if
G
\mathcal {G}
is linear of (permutational) characteristic
0
,
0,
then
G
\mathcal {G}
contains a regular normal abelian subgroup.