Find the subgroup!

1. Dec 6, 2004

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

2. Dec 6, 2004

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

3. Dec 7, 2004

mansi

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

4. Dec 7, 2004

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.

5. Dec 7, 2004

matt grime

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)

6. Dec 12, 2004

mansi

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