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 σ2 = 1.
ii) If G is a group such that o(G) = 2m , where m∈N , then G has a subgroup of order m.
iii) If G = < X > is of order 25, then xα generates G , where α is a factor of 25.
iv) {α1, α2,....αn} is a set only if all the s iα follow a given rule.
v) The characteristic of a field containing (50 − 1) elements is 50.
vi) Every subring of a non-commutative ring is non-commutative.
vii) If (R, ∗1,∗2) is a ring, then ψ : R X R→ R : ψ(r1,r2) = r1 ∗2 r1 is a binary operation.
viii) Not every polynomial that is irreducible over Q[x] is irreducible over Z[x] .
ix) If (D, +,.) is an integral domain such that 1∈ D , then D is a field.
x) The function f , defined by f(x) = x - 1/3 - x , has the same set as domain and as range.
i) False. Even permutations do not necessarily have the property that their square is the identity. For example, the permutation (12) is an even permutation, but (12)^2 = (1)(2) is not the identity.
ii) False. The order of a subgroup must divide the order of the group. If o(G) = 2m, then the only possible subgroups of G are of order 1, 2, m, or 2m. There is no guarantee that G has a subgroup of order m.
iii) True. If G = < X > is a cyclic group of order 25, then any element xα in G generates G, where α is a factor of 25. This is a property of cyclic groups.
iv) The statement is vague
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Show that f d : Q[X]\ {0} → N∪{0} :d(f) = 2degf is a Euclidean valuation on Q[x] .
Consider the ring homomorphism 
. Show that Kerφ = (x,3) . What does the Fundamental Theorem of Homomorphism say in this case?
Let G be a group and H ≤ G . Prove that the only right coset of H in G that is a subgroup of G is H itself.
Give an example, with justification, of a group G with subgroups H and K such that HK is not a subgroup of G .
Give an example, with justification, of each of the following:
i) a zero divisor in Z5 ,
ii) an element of C[0, 1] which is not a zero divisor,
iii) a subring of an integral domain which is not an integral domain.
Give an example, with justification, to show that:
i) the union of integral domains need not be an integral domain;
ii) all finite fields are not isomorphic.
Let G be a group, H∆G and H β≤G/H . Let A = {x∈ G|Hx ∈ β}. Show that
i) A ≤ G , ii) H ∆ A , iii) β = A/H .