Algebra and Geometry | Part IA, 2006

Let θ:GH\theta: G \rightarrow H be a homomorphism between two groups GG and HH. Show that the image of θ,θ(G)\theta, \theta(G), is a subgroup of HH; show also that the kernel of θ,ker(θ)\theta, \operatorname{ker}(\theta), is a normal subgroup of GG.

Show that G/ker(θ)G / \operatorname{ker}(\theta) is isomorphic to θ(G)\theta(G).

Let O(3)O(3) be the group of 3×33 \times 3 real orthogonal matrices and let SO(3)O(3)S O(3) \subset O(3) be the set of orthogonal matrices with determinant 1 . Show that SO(3)S O(3) is a normal subgroup of O(3)O(3) and that O(3)/SO(3)O(3) / S O(3) is isomorphic to the cyclic group of order 2.2 .

Give an example of a homomorphism from O(3)O(3) to SO(3)S O(3) with kernel of order 2.2 .

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