# Answer on Abstract Algebra Question for Jasvinder Singh

Question #44366

Which of the following statements are true and which are false? Justify your answer with a short

proof or a counterexample.

i) On the set f1;2;3g, R = f(1;1); (2;2); (3;3)g is an equivalence relation.

ii) No non-abelian group of order n can have an element of order n.

iii) For every composite natural number n, there is a non-abelian group of order n.

iv) Every Sylow p-subgroup of a finite group is normal.

v) If a commutative ring with unity has zero divisors, it also has nilpotent elements.

vi) If R is a ring with identity and u 2 R is a unit in R, 1+u is not a unit in R.

vii) If a and b are elements of a group G such that o(a) = 2, o(b) = 3, then o(ab) = 6.

viii) Every integral domain is an Euclidean domain.

ix) The quotient field of the ring

fa+ibja;b 2 Zg

is C.

x) The field Q(p2) is not the subfield of any field of characteristic p, where p > 1 is a prime.

proof or a counterexample.

i) On the set f1;2;3g, R = f(1;1); (2;2); (3;3)g is an equivalence relation.

ii) No non-abelian group of order n can have an element of order n.

iii) For every composite natural number n, there is a non-abelian group of order n.

iv) Every Sylow p-subgroup of a finite group is normal.

v) If a commutative ring with unity has zero divisors, it also has nilpotent elements.

vi) If R is a ring with identity and u 2 R is a unit in R, 1+u is not a unit in R.

vii) If a and b are elements of a group G such that o(a) = 2, o(b) = 3, then o(ab) = 6.

viii) Every integral domain is an Euclidean domain.

ix) The quotient field of the ring

fa+ibja;b 2 Zg

is C.

x) The field Q(p2) is not the subfield of any field of characteristic p, where p > 1 is a prime.

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