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Anku posted an Question
May 04, 2021 • 01:33 am 30 points
  • IIT JAM
  • Mathematics (MA)

Pls once show by giving an example..

pls once show by giving an example... . .. . . .. . . . . .

3 Answer(s) Answer Now
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  • comment-profile-img>
    Gattu uday

    (Z,+) is infinite cyclic group , it is generated by -1 and +1 , It is the standard infinite cyclic group

  • comment-profile-img>
    Gattu uday

    Both are not Cyclic groups ...(Z,+) is only infinite cyclic group no other infinite cyclic groups exists

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    there are many infinite groups which are cyclic ....(z,+) is not only single infinite cyclic group

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    But all the Infinite cyclic groups must to isomorphic to (Z,+)

  • Alka gupta best-answer

    no, both are not cyclic groups..... these are only infinite groups

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    Deepak singh 1 Best Answer

    Both are non cyclic group because there does not exist a element which can general all set .

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    sir can u pls show how (z,+) is infinite cyclic group...?..as per me its not cyclic

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    (Z,+ ) is an infinite cyclic group, because all integers can be written by repeatedly adding or subtracting the single number 1. In this group, 1 and −1 are the only generators. And order of 1 and -1 are infinite . So (z,+) is infinite cyclic group.

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    sir but operation is addition then how can we do subtraction..

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    sir pls once tell

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    generators are 1 and -1 . Negative no. generated in this way by -1.

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    sir but definition its becoming wrong know as definition says that there exists an a€G s.t. EACH ELEMENT of G is expressible as x=aⁿ

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    so since 1 is not able to generate all elements of (Z,+) so how can 1 be generator

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    that i am not getting sir..pls once tell..i am confused at that point only..

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    yes , you are correct . (Z,+) is different from others , it has two generators 1 and -1 . 1 generates +ve and -1 generates -ve no.

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    ok sir ..thank u sir

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