... isone -to-< /b> one and < /b> onto, then φ is cal led an < /b> isomorphism. An < /b> isomorphism of agroup with itself is called an < /b> automorphism.Proposition 1.11. Let G and < /b> H be groups,< /b> e1the identity element of G ,and< /b> e2the ... Recall that for A =abcd to < /b> be in O(2), the column vectorsac and< /b> b dmust be unit vectors, and < /b> must be orthogonal.24 2. MATRIX LIE GROUPS< /b> 7. The groups < /b> O(1; 1) and < /b> SO(1; 1). Show ... + b mod n = a + b −n.(Since a and < /b> b are less than n, a +b n is less than n; thus we have closure.) To < /b> showassociativity, note that both (a +b mod n)+c mod n and < /b> a+ (b+ c mod n) mod nare equal to...