Unformatted text preview: injective homomorphisms ' W G ! H since such a homomorphism loses information about G . But in fact, such a homomorphism also reveals cer-tain information about the structure of G that otherwise might be missed. For example, consider the homomorphism from the symmetry group G of the square into the symmetric group S 2 induced by the action of G on the two diagonals of the square, as discussed in Example 2.4.4 . Let N denote the set of symmetries ± of the square such that .±/ D e . You can compute that N D f e;c;d;r 2 g . (Do it now!) From the general theory, which I am about to explain, we see that N is a special sort of subgroup G , called a normal subgroup. Understanding such subgroups helps to un-derstand the structure of G . For now, verify for yourself that N is, in fact, a subgroup....
View Full Document
- Fall '08
- Algebra, Inverse function, Subgroup, 2 g, Homomorphism