Course 111: Algebra, 27th October 2006
1. Consider the unique factorisation theorem, as given in the notes. Assume existence of the factorisation and prove uniqueness.
Hint: the proof is by induction and you should begin by supposing that
a = pα1 1 pα2 2 . . . pαr r = q1β1 q2β2 . . . qsβs
2. Show, writing the Cayley table, that Z4 forms a group under addition.
Is this an abelian or nonabelian group?
3. Recall the example of a group discussed in lectures. G = S3 the group
of all 1-1 mappings of the set x1 , x2 , x3 onto itself. Given φ : S → S
and ψ : S → S such that
φ : x1 → x 2
x2 → x 1
x3 → x 3
and
ψ : x1 → x 2
x2 → x 3
x3 → x 1
Show that (φ ◦ ψ) ◦ (ψ ◦ φ) = ψ and that the list of distinct elements
in G as a result of these mappings is given by e, φ, ψ, ψ 2 , φ ◦ ψ, ψ ◦ φ.