The projective special orthogonal group is the Group obtained from the Special Orthogonal Group on factoring by the Scalar Matrices contained in that Group. In general, this Group is not Simple.
See also Projective Special Linear Group, Projective Special Unitary Group, Special Orthogonal Group
References
Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.
``The Groups , ,
, and
, and .'' §2.4 in
Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England:
Clarendon Press, pp. xi-xii, 1985.