Strongly real elements in finite simple orthogonal groups

Abstract

We prove the strong reality of an infinite series of groups and some elements of a special form in the simple real groups.

Strong reality of finite simple groups

“…By [9, Lemma 10], all semisimple elements of 3 D 4 (q) are strongly real. Moreover, the conjugating involution found in the proof of [9,Lemma 10] satisfies to the following property.…”

“…The first author gratefully acknowledges the support from Deligne 2004 Balzan prize in mathematics, and the Lavrent'ev Young Scientists Competition (No 43 on 04.02.2010) even is proven in [8]. The strong reality of PΩ − 4n (q) for q odd is proven in [9]. Moreover, [10,Theorem 8.5] implies that if q is odd, then PΩ + 4n (q) and Ω 2n+1 (q), together with PΩ − 4n (q), are strongly real if q ≡ 1 (mod 4), while Ω 9 (q) and PΩ + 8 (q) are also strongly real if q ≡ 3 (mod 4).…”

“…The problem of reality and strong reality of finite simple groups and the groups in some sense close to simple was studied by many authors, see [2][3][4][5][6][7][8][9][10][11][12][13]. In particular, the classification of finite simple real groups is obtained in [2].…”

On the Orbital Diameter of Groups of Diagonal Type

“…A group is strongly real if and only if any of its elements can be expressed as a product of two involutions. Theorem 2.3 ([8,10,12,13,17,23,25,27,28]). Let G be a non-abelian finite simple group.…”

Strongly real elements of orthogonal groups in even characteristic

“…The alternating group A n is strongly real if and only if n A f5; 6; 10; 14g (see [1]), and it has been shown in [3], [5], [6] that PSp 2n ðqÞ is strongly real if q 2 3 ðmod 4Þ. Knü ppel and Thomsen [7] have determined which of the orthogonal groups in odd characteristic are strongly real, and Galt [4] has independently obtained the same results for some of the orthogonal groups. Kolesnikov and Nuzhin [8] have shown that of the sporadic groups, only J 1 and J 2 are strongly real.…”