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
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Show that f d : Q[X]\ {0} → N∪{0} :d(f) = 2degf is a Euclidean valuation on Q[x] .
How many Sylow 5-subgroups, Sylow 3-subgroups and Sylow 2-subgroups can a group of order 200 have? Give reasons for your answers.
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 .
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Prove that if G ≠ {e} and G has no proper non-trivial subgroup, then G is finite and O(G) is a prime number.
Let p be a prime. Show that S = {m + sqrt(-p)|m, n∈Z} is a subring of C. Further, check whether or not S is an ideal of C.