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  1. #1
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    question

    P and q are two distinct primes. H ia s proper subset of integers and also it is a group under addition. another set A={p,p+q,pq,p^q,q^p}. now H has three elements and those three elements are from A. which are those three elements? please reply in detail
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by luckylawrance View Post
    P and q are two distinct primes. H ia s proper subset of integers and also it is a group under addition. another set A={p,p+q,pq,p^q,q^p}. now H has three elements and those three elements are from A. which are those three elements? please reply in detail
    I presume you mean H \cap A contains 3 elements, not H contains three elements (as subgroups of \mathbb{Z} are either trivial or contain infinitely many elements).

    Now, subgroups of \mathbb{Z} are all of the form n\mathbb{Z} = \{ni; i \in \mathbb{Z}\} (why?). Thus, there exists n \in \mathbb{Z} such that H=<n> ( H is the subgroup generated by n).

    Basically, this all translates as "the elements you are looking for are elements a, b, c \in A such that gcd(a, b, c) \neq 1" (not equal to 1 as H is a proper subgroup).

    Can you find three such elements?
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  3. #3
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    first of all thank you for replying. but question in the book goes like this only..that's why i am unable to get the solution
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by luckylawrance View Post
    first of all thank you for replying. but question in the book goes like this only..that's why i am unable to get the solution
    Well, what bit don't you understand?
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  5. #5
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    the solution given in book is {p,p+q,p^q}. but how is it possible that sum of any two of these three elements also lies in the group?
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  6. #6
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by luckylawrance View Post
    the solution given in book is {p,p+q,p^q}. but how is it possible that sum of any two of these three elements also lies in the group?
    I believe it should be pq, not p+q (if my interpretation of the question, which I posted earlier, is correct).
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