63Solution.We use Exercise 2.69:Qhas exactly one element of order 2,whileD8has 5 elements of order 2.2.88IfGis afnite group generated by two elements of order 2, prove thatG∼=D2nfor somen≥2.Solution.Absent.2.89(i)Prove thatA3is the only subgroup ofS3of order 3.Solution.Absent.(ii)Prove thatA4is the only subgroup ofS4of order 12. (In Exer-cise 2.135, this will be generalized fromS4andA4toSnandAnfor alln≥3.)Solution.IfHis a subgroup of order 12, thenHis normal (ithas index 2), and so it contains all the conjugates of each of itselements. It must contain 1. We count the number of conjugates ofthe various types of permutations (each count uses Exercise 2.24:(12)has 6 conjugates;(123)has 8 conjugates;(1234)has 6conjugates;()(34)has 3 conjugates. The only way to get 12elements is 1+3+8; but this isA4.2.90(i)LetAbe the set of all 2×2 matrices of the formA=£ab01±, wherea±=0. Prove thatAis a subgroup of GL(2,R).Solution.It is a routine calculation to show that if
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GL, φ, Group isomorphism, conjugates, afﬁne group Aff