Geodesic
ON THE NUMBER OF REAL CLASSES IN THE FINITE PROJECTIVE LINEAR AND UNITARY GROUPS
Glasgow mathematical journal, Tome 62 (2020) no. 1, pp. 93-107

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We show that for any n and q, the number of real conjugacy classes in $ \rm{PGL}(\it{n},\mathbb{F}_q) $ is equal to the number of real conjugacy classes of $ \rm{GL}(\it{n},\mathbb{F}_q) $ which are contained in $ \rm{SL}(\it{n},\mathbb{F}_q) $, refining a result of Lehrer [J. Algebra36(2) (1975), 278–286] and extending the result of Gill and Singh [J. Group Theory14(3) (2011), 461–489] that this holds when n is odd or q is even. Further, we show that this quantity is equal to the number of real conjugacy classes in $ \rm{PGU}(\it{n},\mathbb{F}_q) $, and equal to the number of real conjugacy classes of $ \rm{U}(\it{n},\mathbb{F}_q) $ which are contained in $ \rm{SU}(\it{n},\mathbb{F}_q) $, refining results of Gow [Linear Algebra Appl.41 (1981), 175–181] and Macdonald [Bull. Austral. Math. Soc.23(1) (1981), 23–48]. We also give a generating function for this common quantity. 
AMPARO, ELENA; VINROOT, C. RYAN. ON THE NUMBER OF REAL CLASSES IN THE FINITE PROJECTIVE LINEAR AND UNITARY GROUPS. Glasgow mathematical journal, Tome 62 (2020) no. 1, pp. 93-107. doi: 10.1017/S0017089518000551
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     title = {ON {THE} {NUMBER} {OF} {REAL} {CLASSES} {IN} {THE} {FINITE} {PROJECTIVE} {LINEAR} {AND} {UNITARY} {GROUPS}},
     journal = {Glasgow mathematical journal},
     pages = {93--107},
     year = {2020},
     volume = {62},
     number = {1},
     doi = {10.1017/S0017089518000551},
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