Strongly real classes in finite unitary groups of odd characteristic

Abstract

We classify all strongly real conjugacy classes of the finite unitary group U(n, F q ) when q is odd. In particular, we show that g ∈ U(n, F q ) is strongly real if and only if g is an element of some embedded orthogonal group O ± (n, F q ). Equivalently, g is strongly real in U(n, F q ) if and only if g is real and every elementary divisor of g of the form (t ± 1) 2m has even multiplicity. We apply this to obtain partial results on strongly real classes in the finite symplectic group Sp(2n, F q ), q odd, and a generating function for the number of strongly real classes in U(n, F q ), q odd, and we also give partial results on strongly real classes in U(n, F q ) when q is even.2010 Mathematics Subject Classification: 20G40, 20E45

Real classes of finite special unitary groups

“…It is known that every real class of GL n (q) is also strongly real, which was first shown for odd q by Wonenburger [21] and in general by by Djoković [4] (these statements are also proved independently in [8]). This statement is far from true in the group GU n (q), as given by the following main result of [7].…”

“…This statement is far from true in the group GU n (q), as given by the following main result of [7]. Theorem 2.1 (Gates, Singh, and Vinroot).…”

“…It follows from the work of Wall [20, Sec. 2.6, Case (A)] (see also [7,Proposition 2.1]), that for g * = ⊕ f =t±1 g f , we have…”

“…Gill and Singh [8,9] have classified all of the real and strongly real classes of the finite special and projective linear groups, as well as all quasi-simple covers of the finite projective special linear groups. Gates, Singh, and the second-named author of this paper [7] classified the strongly real classes of the finite unitary group.…”

The involution width of finite simple groups

“…Clearly the elementary divisor (t − 1) is also self-reciprocal and it follows that x is real in SU n (q). It then follows from ([49], 7.1) and ( [20], 2.2 and 2.4) that x is strongly real in SU n (q). A similar method is used for y where we now pick a regular semisimple element in the torus 2 .…”