Conjugacy classes of small sizes in the linear and unitary groups

Abstract

Abstract. Using the classical results of G. E. Wall on the parametrization and sizes of (conjugacy) classes of finite classical groups, we present some gap results for the class sizes of the general linear groups and general unitary groups as well as their variations. In particular, we identify the classes in GL n .q/ of size up to q 4n 8 and classes in GU n .q/ of size up to q 4n 9 . We then apply these gap results to obtain some bounds and limits concerning the zeta-type function encoding the conjugacy class sizes of these groups.

Variations of Landau's theorem for p-regular and p-singular conjugacy classes

“…These bounds are probably not the best possible but they are enough for our purpose. We basically make use of a result of L. Babai, P. P. Pálfy, and J. Saxl [2] on the proportion of p-regular elements in finite simple groups and a recent result of J. Fulman and R. M. Guralnick on centralizer sizes in finite classical groups, see [9,5]. The detailed structure of the centralizers of semisimple elements in finite classical groups can be found in [28].…”

Equivariant Euler characteristics of partition posets

“…(1 + q 4 x)((1 + q 2 x) 6 (1 + x) (1 − q 3 x) 4 (1 − qx) 4 for r + 1 = 1, 2, 3, 4. In particular, − χ 2 (GL − n (F q )) = q + 1 and − χ 3 (GL − n (F q )) = nq n−1 (q + 1) 2 for n ≥ 1.…”