Answer to Question #152840 in Abstract Algebra for Sourav Mondal

Question

Which of the following statements are true, and which short proof or a counter-example.

i)There is no non-abelian group of 12

ii)If in a group G every element is of finite order, then G is a finite order.

iii)The homomorphic image of a non-cyclic group is non cyclic.

iv) If a is an integral domain, then R /I is an integral domain for every non-zero ideal I of R .

v)If I and J are ideals of a ring R, then so is I U J

Answer 1

i) False $A_{6}$ alternating group is a non abelian group of order 12.

ii) False: $\mathbb{Q}/\mathbb{Z}$ is an infinite group with every element of finite order.

iii) False $\phi:S_3\longrightarrow \mathbb{Z_2}$ given by $\phi((123))=\phi((132))=\phi(e)=\overline{0}.$$\phi((12))=\phi((13))=\phi((23))=\overline{1}.$ This is a homomorphism . The homomorphic image of non cyclic $S_3$ is the cyclic group $\mathbb{Z}_{2}.$

(iv) False. Take R=$\mathbb{Z}$ and I=$6\mathbb{Z}$ . $R/I\cong \mathbb{Z}_{6}$ not an integral domain.

(v) False. Take R=$\mathbb{Z}$, I=$2\mathbb{Z},$ J=$3\mathbb{Z}.$ But $I\cup J$ not ideal, since $5\notin I\cup J$ but $2,3\in I\cup J$ . So for an ideal $2+3=5$ must belong to it.