Extending real-valued characters of finite general linear and unitary groups on elements related to regular unipotents

Abstract

Let GLðn;

On the maximum orders of elements of finite almost simple groups and primitive permutation groups

“…AS type HS or SD PA type type Alt(5) M 11 PSL 2 (7) PSL 2 (49) PSU 3 (3) PSp 6 (2) Alt(5) 2 T ℓ where Alt(6) M 12 PSL 2 (8) PSL 3 (3) PSU 3 (5) PSp 8 (2) T is one of Alt(7) M 22 PSL 2 (11) PSL 3 (4) PSU 4 (3) PSp 4 (3) the groups Alt(8) M 23 PSL 2 (16) PSL 4 (3) in the AS type Alt(9) M 24 PSL 2 (19) part of HS PSL 2 (25) this table Table 2. The socles for the exceptions G in Theorem 1.3 (4) 1.1.…”

On orders of elements of finite almost simple groups with linear or unitary socle

“…Following the lines of the proof of [20, Proposition 13], we will exploit a correspondence between σ-conjugacy classes of C K (σ k ) and conjugacy classes of C K (σ), where K is a connected linear algebraic group and σ is a Steiberg endomorphism of K, i. e. a surjective endomorphism with finitely many fixed points. Also we will use a slight modification of this correspondence inspired by [9,Theorem 2.1].…”

Generating functions for real character degree sums of finite general linear and unitary groups

“…It is known, by [2, 1.1.1] and [8,Lemma 3.1], that a character of GL(n, q) is real precisely when λ(φ) = λ(φ * ) for all monic irreducible polynomials φ. Thus the sum of the degrees of the real characters of GL(n, q) is equal to…”

“…By Theorem 3.2, the sum of the degrees of the real-valued characters of GL(n, q), when q is odd, is (q n − 1) · · · (q − 1) times the coefficient of u n in the left side of (8). This, together with (8), gives that this sum of character degrees is the number of elements in GL(n, q), q odd, which square to the identity.…”