Let p be a prime and a ∈ N such that p|a⁵⁰ . Show that p⁵⁰|a⁵⁰ .

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  x^α generates G , where α is a factor of 25.

iv) {α₁, α₂,....α_n}  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.
 

Prove that an element of an integral domain is a unit iff it generates the domain.

Prove that Rⁿ/R^m ~ R^{n - m} as groups, where  n,m ∈N, n ≥ m .

Prove that if G ≠ {e} and G has no proper non-trivial subgroup, then G is finite and O(G) is a prime number.

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.

Check whether f :(4Z, +) → ( Z₄, +) : f (4m) = is a group homomorphism or not. If it is, what does the Fundamental Theorem of Homomorphism give us in this case? If f is not a homomorphism, obtain the range of f .