Math Answers

Let K be a field and f : Z → K the homomorphism of

integers into K.

a) Show that the kernel of f is a prime ideal. If f is an embedding,

then we say that K has characteristic zero.

b) If kerf f= {0}, show that kerf is generated by a prime number

p. In this case we say that K has characteristic p.

A factory employer claims that their company follows the hours

of work for its employees and the standard deviation of their

working time is 0.5 hours.

It is given that the total marks of all probability students have an average of 45 marks with a standard deviation of 15 marks. If a student from a sample of 9 students is chosen at random, what is the probability that his marks are between 40 and 60?

2.10. Let H be the subgroup generated by two elements a, b of a group G. Prove that if ab = ba, then H is an abelian group.

2.9. Let a and b be integers.

(a) Prove that the subset aZ + bZ = {ak + bl | l, k ∈ Z } is a subgroup of Z.

(b) Prove that a and b + 7a generate the subgroup aZ + bZ.

2.8. Let a, b be elements of a group G. Assume that a has order 5 and a³b = ba³. Prove that ab = ba.

2.7. If G is a group such that (ab)2 = a2b2 for all a, b ∈ G, then show that G must be abelian.

2.6. If G is a group in which (ab)ⁱ = aⁱbⁱ for three consecutive integers i for all a, b ∈ G, show that G is abelian.

2.5. If G is a finite group, show that there exists a positive integer m such that a^m = e for all a ∈ G.

2.4. If G is a group of even order, prove that it has an element "a\\ne e" satisfying a² = e.