Answer to Question #152840 in Abstract Algebra for Sourav Mondal

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 "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.


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