Affiliation:
1. University of the Highlands and Islands , Inverness , United Kingdom
2. University of Manchester , Manchester , United Kingdom
Abstract
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.
Subject
Algebra and Number Theory
Reference17 articles.
1. M. Aschbacher and G. M. Seitz,
Involutions in Chevalley groups over fields of even order,
Nagoya Math. J. 63 (1976), 1–91.
2. J. Ballantyne,
Local fusion graphs of finite groups,
PhD. Thesis, University of Manchester, 2011.
3. J. Ballantyne, N. Greer and P. Rowley,
Local fusion graphs for symmetric groups,
J. Group Theory 16 (2013), no. 1, 35–49.
4. J. Ballantyne and P. Rowley,
A note on computing involution centralizers,
J. Symbolic Comput. 54 (2013), 1–8.
5. J. Ballantyne and P. Rowley,
Local fusion graphs and sporadic simple groups,
Electron. J. Combin. 22 (2015), no. 3, Paper 3.18.