American Journal of Applied Sciences 5 (9): 1107-1109, 2008 ISSN 1546-9239 © 2008 Science Publications Current Address: Ibrahim A.I. Suleiman, Department of Mathematical and Physical Sciences, University of Nizwa, Birkat ALMouz, P.O. Box 33, 616 Nizwa, Sultanate of Oman 1107 Strongly Real Elements in the Linear Groups PSL n (q) Ibrahim A.I. Suleiman Department of Mathematics and Statistics, Mu’tah University, P.O.Box (7), 61710 Al-Karak, Jordan Abstract: We investigated which elements in some projective special linear and unitary groups are strongly real. In particular we showed every real element in PSL 2 (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. Key words: Real elements, linear groups 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. Similarly the conjugacy class g G is called real or strongly real if g is. Clearly every strongly real element is real, but the converse is not true. For example, the quaternion group Q 8 has the property that all its elements are real, but its only involution is central, so cannot conjugate an element of order 4 to its inverse. In a recent study [2] , Tiep and Zalesski determine which finite simple groups have the property that every element is real. 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 a preprint [7] , it has been determined exactly which elements in the sporadic and the alternating groups are strongly real. 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 J 1 and J 2 . It is known [8,9] ) that all real elements in the general linear group GL n (q) are strongly real. However, this is not true in the special linear groups SL n (q) for example, if n = 2, then there is only one involution in SL 2 (q), so only the central elements of SL 2 (q) are strongly real. On the other hand, we prove that all real elements in PSL 2 (q) are strongly real. Indeed, we conjecture that all real elements in PSL n (q) are strongly real, for arbitrary n. Moreover, we completely classify the strongly real elements in many unitary groups as far as they are in [1] . We used GAP [5] to find all elements in all these groups which are real but not strongly real. GENERAL LINEAR GROUPS The general linear group GL n (q) consists of all n×n matrices with entries in F q . Equivalently it is the group of all linear automorphisms of an n-dimensional vector space over F q . The special linear group SL n (q) is the subgroup of all matrices of determinant 1. On factoring GL n (q) and SL n (q) by the scalar matrices we get what is known as the projective general linear group PGL n (q) and projective special linear group PSL n (q). The orders of these groups are given by the following formulae: n n n GL (q) (q 1)N, SL (q) PGL (q) N = - = = and n n N PL (q) L (q) d = = where, n(n 1) / 2 n 2 N q (q 1) .... (q 1) - = - - and d is the greatest common divisor of q-1 and n. Projective special linear groups ln (q): We know from [2] , that the only groups L n (q) in which all elements are real are the groups L n (q) with either q = 2 k or q ≡1 mod 4. Here we show that in fact all