Strongly Real Elements in the Linear Groups PSLn(q)

Ibrahim A.I Suleiman

doi:10.3844/ajassp.2008.1107.1109

Abstract

We investigated which elements in some projective special linear and unitary groups are strongly real. In particular we showed every real element in PSL2(q) strongly real. We write a full table for real classes which are not strongly real in the unitary groups as well as a table for the non real classes in the unitary groups.

Highlights

An element g in a group G is called real if g is conjugate to g−1 and is called strongly real if it is the product of two involutions
We prove that all real elements in PSL2 (q) are strongly real
We conjecture that all real elements in PSLn (q) are strongly real, for arbitrary n

Summary

INTRODUCTION

An element g in a group G is called real if g is conjugate to g−1 and is called strongly real if it is the product of two involutions. A much more difficult question, raised in problem 14.82 of the Kourovka notebook[6], is which finite simple groups have the property that every element is strongly real. This problem is still open in general (but see[3,4], for some cases, and related questions). In the alternating groups every real element is strongly real, but this is not true in all sporadic groups. The alternating and the sporadic groups in which every element is strongly real are An for (n = 5, 6, 10, 14), and J1and J2. We used GAP[5] to find all elements in all these groups which are real but not strongly real

GENERAL LINEAR GROUPS

Because q