... one -to- one and onto, then φ is called an isomorphism An isomorphism of a group with itself is called an automorphism Proposition 1.11 Let G and H be groups, e1 the identity element of G, and ... are non-compact: O(n; C) and SO(n; C); O(n; k) and SO(n; k) (n ≥ 1, k ≥ 1); the Heisenberg group H; Sp (n; R) and Sp (n; C); E(n) and P(n; 1); R and Rn ; R∗ and C∗ It is left to the reader to provide ... Rudin, Real and Complex Analysis, Chap 9, Ex 17.) If G and H are matrix Lie groups, and there exists a Lie group isomorphism from G to H, then G and H are said to be isomorphic, and we write...