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Which of the following statements are true? Give reasons for your answers. Marks will only be given for valid reasons. i) If σ is an even permutation, then I 2 σ = . ii) If G is a group such that m (o G) = 2 , where m∈N , then G has a subgroup of order m. iii) If G is of order 25, then = < x > α x generates G , where α is a factor of 25. iv) { , , , } α1 α2 K αn is a set only if all the s αi follow a given rule. v) The characteristic of a field containing )1 50( − elements is 50. vi) Every subring of a non-commutative ring is non-commutative. vii) If ) ( ,R , 1 2 ∗ ∗ is a ring, then 1 2 1 2 1 ψ : R × R → R :ψ(r r, ) = r ∗ r is a binary operation. viii) Not every polynomial that is irreducible over ]x[ Q is irreducible over ]x[ Z . ix) If .), ( ,D + is an integral domain such that 1∈D , then D is a field. x) The function f , defined by 3 x x 1 )x(f − − = , has the same set as domain and as range.
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