Abstract
Let 𝐺 be isomorphic to
GL
n
(
q
)
\mathrm{GL}_{n}(q)
,
SL
n
(
q
)
\mathrm{SL}_{n}(q)
,
PGL
n
(
q
)
\mathrm{PGL}_{n}(q)
or
PSL
n
(
q
)
\mathrm{PSL}_{n}(q)
, where
q
=
2
a
q=2^{a}
.
If 𝑡 is an involution lying in a 𝐺-conjugacy class 𝑋, then, for arbitrary 𝑛, we show that, as 𝑞 becomes large, the proportion of elements of 𝑋 which have odd order product with 𝑡 tends to 1.
Furthermore, for 𝑛 at most 4, we give formulae for the number of elements in 𝑋 which have odd order product with 𝑡, in terms of 𝑞.
Large orbits of odd-order subgroups of solvable linear groups
