Course 111: Algebra, 28nd November 2006
1. Prove that for a homomorphism φ : G → Ḡ: (i) φ(e) = ē and (ii)
φ(x−1 ) = φ(x)−1 .
Proof (i) Calculate φ(x)ē = φ(x) = φ(xe) = φ(x)φ(e) and by the
cancellation property in Ḡ we have that φ(e) = ē as required.
Proof (ii) Note that ē = φ(e) = φ(xx−1 ) = φ(x)φ(x−1 ) so by the
definition of φ(x)−1 in Ḡ we obtain the result that 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.
All elements of the Klein 4-group have order 2 except for the identity
which has order 1.
The cyclic group, C4 has 1 element of order 1, 1 element of order 2 and
2 elements of order 4.
Consider an element of order 4 in C4 , written a and a possible homomorphism or isomorphism from C4 to V4 . Then, φ(a.a.a.a) = φ(e) since
a of order 4. But now φ(a).φ(a).φ(a).φ(a) 6= ē since there is no element
of order 4 in V4 and so we have φ(a.a.a.a) 6= φ(a).φ(a).φ(a).φ(a) and
so there is no map satisfying the properties of an isomorphism from C4
to V4 .
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.
Proof
φ(xy) = log10 (xy)
= log10 (x) + log10 (y)
= φ(x) + φ(y)
proving that φ is a homomorphism. To prove φ is a bijection we just
need to find a corresponding inverse map from Ḡ → G. This map is
φ̄(y) = 10x for each x ∈ G and y ∈ Ḡ.