The automorphism group of $$NU(3,q^2)$$

Abstract

AbstractLet $$H(n, q^2)$$ H ( n , q 2 ) be a non-degenerate Hermitian variety of $$PG(n,q^2)$$ P G ( n , q 2 ) , $$n \ge 2$$ n ≥ 2 . Let $$NU(n+1,q^2)$$ N U ( n + 1 , q 2 ) be the graph whose vertices are the points of $$PG(n,q^2) \setminus H(n,q^2)$$ P G ( n , q 2 ) \ H ( n , q 2 ) and two vertices u,  v are adjacent if the line joining u and v is tangent to $$H(n, q^2 )$$ H ( n , q 2 ) . Then $$NU(n + 1, q^2)$$ N U ( n + 1 , q 2 ) is a strongly regular graph. In this paper we show that the automorphism group of the graph $$NU(3,q^2)$$ N U ( 3 , q 2 ) is isomorphic either to $$P\Gamma U(3,q)$$ P Γ U ( 3 , q ) , the automorphism group of the projective unitary group PGU(3, q), or to $$S_{3} \wr S_4$$ S 3 ≀ S 4 , according as $$q \ne 2$$ q ≠ 2 , or $$q=2$$ q = 2 .