M2PM2 Algebra II, Exam 2010
1. (a) (i) Let x∈Snbe a product of disjoint cycles of lengths r1, r2,...,rm.
Write down a condition on r1,...,rmfor xto be an even permutation.
(ii) Define the alternating group An.
(iii) For which values of nis Anabelian ? Justify your answer.
(iv) Calculate the number of elements of order 2 in the group A6.
(v) Find the smallest value of nsuch that Ancontains an element of order
60. Justify your answer.
(vi) Prove that if x∈Snis an odd permutation, then Sn=An∪Anx.
(b) Calculate the number of distinguishable ways there are of colouring the
edges of a square using 3 colours, if each colour can be used any number of
times.
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