Course 111: Algebra, 28nd November 2006
To be handed in at tutorials on Dec 4th and 5th.
1. Prove that for a homomorphism φ : G → Ḡ: (i) φ(e) = ē and (ii)
φ(x−1 ) = φ(x)−1 .
2. Consider the Klein 4-group, V4 whose Cayley table was given in the
notes (when we considered an example of quotient groups) and the
cyclic group of four elements, G = {e, g, g 2, g 3 }.
Determine the order of the elements in these groups and use this information to show that the groups are not isomorphic.
3. Let G be the group of positive real numbers under multiplication and
let Ḡ be the group of all real numbers under addition. Define a map
φ : G → Ḡ by φ(x) = log10 x. Prove that φ is a homomorphism and in
fact that φ is an isomorphism.