Which 3 elements from {p, p+q, pq, p^q, q^p} form a group under addition?

[mansi]

let p and q be distinct primes. suppose that H is a proper subset of integers and H is a group under addition that contains exactly 3 elements of the set
{ p,p+q,pq, p^q , q^p}.
Determine which of the foll are the 3 elements in H
a. pq, p^q, q^p

b. P+q, pq,p^q

c. p, p+q, pq

d. p, p^q, q^p

e. p, pq, p^q

 

[matt grime]

Hint Euclid's algorithm: p^r and q^s are coprime so if H contains these two elements, then it contains 1, and hence is Z. Use this idea in several variations. Of course you could consider the group pZ

 

[mansi]

thanks sir, but could you please elaborate further.
i don't seem to get the idea...

 

[matt grime]

If a group contains p, it contains np for all n in Z. So clearly e. forms the answer.

A group for instance cannot contain p and q if they are coprime and not be all of Z since there are a and b in Z such that ap+bq=1, hence the group contains all elements of Z.

And I tihnk you ought to ponder that for a while, cos I really have given you more information than I want to.

 

[matt grime]

How about thinking about an example if you cant' see it:

p=2 q=3

If 2 and 3 are in the group, then so is -2 (inverses) and hence, so is 3-2=1 (composition)

If 1 is in there so is 1+1+1+..+1= n (composition) and n was arbitrary, also -n is in there (inverses again)

 

[mansi]

well...thanks a lot, sir! i figured it out...