Thread: 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

Originally Posted by luckylawrance

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 contains 3 elements, not contains three elements (as subgroups of are either trivial or contain infinitely many elements).

Now, subgroups of are all of the form (why?). Thus, there exists such that ( is the subgroup generated by ).

Basically, this all translates as "the elements you are looking for are elements such that " (not equal to 1 as H is a proper subgroup).

Can you find three such elements?

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

Originally Posted by luckylawrance

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?

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?

Originally Posted by luckylawrance

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).