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Definition: External Direct Products
Let G and K be two groups. Define GxK by
GxK= { (g,k) I gEG, kEK }
with the operation * defined on GxK by
(g1,k1) * (g2,k2) = (g1g2,k1k2) for g1,g2 EG, k1,k2EK.
where the operation in the first coordinate is the operation of G and the operation in the second coordinate is the operation of K.
We say that GxK is the external direct product of G and K.
Questions:
1. Prove that this algebraic structure (Gxk,*) gives us a group.
2. Suppose G = K = Z2 ,
a) List all the elements of GxK ,
b) Is GxK cyclic?
c) Show that GxK is isomorphic to the Klein four-group, V.
3. For each pair G and K specified below:
i) G = Ζ3 and K = Ζ2
ii) G = Ζ4 and K = Ζ4
iii) G = Ζ3 and K = Ζ4
iv) G = Ζ6 and K = Ζ2
a) Write out all the elements of GxK .
b) Make a conjecture about the order of GxK when G and K are finite groups. Prove your conjecture.
4. Refer to question 3,
a) From a)(i) and a)(ii) in question no.3, determine the order of each element (g,k)EGxK and also determine and the order of gEG and kEK.
b) What is the relationship between the order of element (g,k)EGxK and the order of gEG and kEK. Make a conjecture about it if G and K are finite. Justify your answer.
c) From each of the direct product in question 3, is GxK cyclic? Make a conjecture about the structure of cyclic groups of GxK if G and K are finite. Justify your answer.