Question #152840

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


1
Expert's answer
2020-12-26T14:53:18-0500


i) False A6A_{6} alternating group is a non abelian group of order 12.

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

iii) False ϕ:S3Z2\phi:S_3\longrightarrow \mathbb{Z_2} given by ϕ((123))=ϕ((132))=ϕ(e)=0.\phi((123))=\phi((132))=\phi(e)=\overline{0}.ϕ((12))=ϕ((13))=ϕ((23))=1.\phi((12))=\phi((13))=\phi((23))=\overline{1}. This is a homomorphism . The homomorphic image of non cyclic S3S_3 is the cyclic group Z2.\mathbb{Z}_{2}.

(iv) False. Take R=Z\mathbb{Z} and I=6Z6\mathbb{Z} . R/IZ6R/I\cong \mathbb{Z}_{6} not an integral domain.

(v) False. Take R=Z\mathbb{Z}, I=2Z,2\mathbb{Z}, J=3Z.3\mathbb{Z}. But IJI\cup J not ideal, since 5IJ5\notin I\cup J but 2,3IJ2,3\in I\cup J . So for an ideal 2+3=52+3=5 must belong to it.


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