Question

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.

Answer 1

Answer on Question #44366 – Math - Abstract Algebra

Problem.

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 $u 2R$ 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

$$f a + i b j a; b 2 Z g$$

is $C$.

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

**Remark.** The statement of the problem is formatted incorrectly. I suppose that the correct statement is

"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 $\{1,2,3\}$, $R = \{(1;1), (2;2), (3;3)\}$ 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 $u \in R$ 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

$$\{a + i b \mid a, b \in \mathbb {Z} \}$$

is $\mathbb{C}$.

x) The field $\mathbb{Q}(\sqrt{2})$ is not the subfield of any field of characteristic $p$, where $p > 1$ is a prime."

Solution.

i) True.

The reflexivity holds, as for all $\in \{1,2,3\} (x,x)\in R$.

The symmetry holds, as if $(x,y)\in R$, then $(y,x)\in R$, as there are only elements of type $(x,x)$ in $R$.

The transitivity holds, as if $(x,y) \in R$, $(y,z) \in R$, then $(x,z) \in R$, as there are only elements of type $(x,x)$ in $R$.

ii) True.

If there is element of order $n$ in the group of order $n$, then this group is cyclic. The cyclic group is abelian.

iii) False.

Each group of order 4 is isomorphic to $\mathbb{Z}_4$ or to $\mathbb{Z}_2 \times \mathbb{Z}_2$.

iv) False.

The $S_4$ doesn't have normal Sylow subgroup.

v) False.

$\mathbb{Z}_6$ has zero divisor, but doesn't have nilpotent elements.

vi) False.

If $R = \mathbb{R}$ and $u = 2$, then $u = 3$ is unit.

vii) False.

If $a = (1\ 2) \in S_3$ and $b = (1\ 2\ 3) \in S_3$, then $ab = (3\ 2) \in S_3$ and $o\big((3\ 2)\big) = 2$.

viii) False.

$\mathbb{Q}(\sqrt{-19})$ is integral domain, but isn't an Euclidean domain.

ix) False.

The quotient field is field with element $(x, y)$ where $x, y \in R$ and $R$ should be an integral domain.

x) True.

If field $\mathbb{Q}(\sqrt{2})$ is the subfield of any field of characteristic $p$, then $\sqrt{2}p = 0$.

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