64Solution.It is easy to see thatϕis a homomorphism:ϕ(MN)=QMNQ−1=QMQ−1QNQ−1=M)ϕ(N).Next, ifM=·abcd¸,thena+c=1 andb+d=1. It follows that imϕ≤A, for matrixmultiplication gives·1011¸·−¸=·a−bb01¸(the bottom row of the last matrix is(a+c)−(b+d)b+d,and this is 0 1 becauseMis stochastic.Finally, we must show thatϕis surjective. This is obvious fromthe last calculation. IfA=·¸∈A,thenA=M), whereM=·a+1−a−b1−b¸.2.91Prove that the symmetry group6(πn), whereπnis a regular polygon withnvertices, is isomorphic to a subgroup ofSn.Solution.Every isometryϕ∈n)permutes thenverticesX
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.