Abstract Algebra group problem.

In summary, the problem is trying to find a solution to the equation p^q = 1 that does not have p in the equation.
  • #1
WHOAguitarninja
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ProbelmLet p and q be distinct primes. Suppose that H is a porper subset of the integers and H is a group under addition that contains exactly three elements of the set {p, p+q, pq, p^q, q^p}. Determine which of the following are the three 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

Homework Equations


The answer is e, as it's given in the book.

The Attempt at a Solution


My basic strategy here was to try and show it's only closed under addition for e. It's easy to see that adding p to itself q times yields pq, however the same cannot be said for p^q. This didn't seem satisfactory then. My next idea, the one that MIGHT be right, but I'm very much not sure, is that e contains the only terms that are integer factors of p. You know p, pq, and p^q are integers when divided by p, and furthermore they are the only ones that are (since p and q are prime). However, I don't know if this is even signficant, and I certainly don't see the signifigance under a group who's operation is addition.

EDIT - Actually I suppose it's significant in that being an integer multiple it can be seen as "P+P+P+..." an integer number of times. Is that all there is to it? It seems a bit of an odd problem if so.
 
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  • #2
Hint 1: H has to be a proper subset of Z
Hint 2: If x and y are coprime, then there exists integers a and b such that ax + by = gcd(x,y)
 
  • #3
I've been fiddling for a little while and can't seem to come up with anything satisfactory. The crux of the problem seems to be that the items in the group must be integers, however I can't seem to find the relevance of your second hint. Having fiddled with it, it doesn't seem to get me anywhere.

Is it not important that only the terms in e can have p factored out and have the resulting term still be integer valued?
 
  • #4
WHOAguitarninja said:
I've been fiddling for a little while and can't seem to come up with anything satisfactory. The crux of the problem seems to be that the items in the group must be integers, however I can't seem to find the relevance of your second hint. Having fiddled with it, it doesn't seem to get me anywhere.

Is it not important that only the terms in e can have p factored out and have the resulting term still be integer valued?
Yes, that's good. Since p, pq, and pq are integer multiples of p, they can appear in the subgroup of Z consisting of all integer multiples of p, which is indeed a proper subgroup. Now with all the other options, a-d, you can't factor out p from each of the three elements. So p is never a common factor of the elements in a-d. But you need to show that for each of a-d, the three elements have no common factors at all.

The point of my hint was that if in each of a-d, you can find a pair of elements that are coprime, e.g. in b, pq and p+q are coprime. Therefore, there exist integers a and b such that a(pq) + b(p+q) = 1. But then 1 would be in H, and hence H would be all of Z, contradicting Hint 1.
 
  • #5
AKG said:
Yes, that's good. Since p, pq, and pq are integer multiples of p, they can appear in the subgroup of Z consisting of all integer multiples of p, which is indeed a proper subgroup. Now with all the other options, a-d, you can't factor out p from each of the three elements. So p is never a common factor of the elements in a-d. But you need to show that for each of a-d, the three elements have no common factors at all.

The point of my hint was that if in each of a-d, you can find a pair of elements that are coprime, e.g. in b, pq and p+q are coprime. Therefore, there exist integers a and b such that a(pq) + b(p+q) = 1. But then 1 would be in H, and hence H would be all of Z, contradicting Hint 1.


Thanks. That makes perfect sense. I wasn't thinking more along the lines of showing a-d were not groups than were not subgroups. I suppose it then makes sense that I could never explain that they weren't, because they are! Thanks alot, that really helped.
 

What is abstract algebra and how is it related to group problems?

Abstract algebra is a branch of mathematics that studies algebraic structures, such as groups, rings, fields, and vector spaces. Group problems are problems that involve the properties and operations of groups, which are one of the fundamental structures studied in abstract algebra.

What is a group and what are its basic properties?

A group is a set of elements with a binary operation (such as addition or multiplication) that satisfies four basic properties: closure, associativity, identity, and invertibility. Closure means that the result of the operation on any two elements of the group is also an element of the group. Associativity means that the order in which the operations are performed does not matter. Identity means that there is an element in the group that when operated on with any other element, returns that element unchanged. Invertibility means that every element in the group has an inverse element that, when operated on together, results in the identity element.

How do you determine if a set is a group?

To determine if a set is a group, you must check if the set satisfies the four basic properties of a group: closure, associativity, identity, and invertibility. If all four properties are satisfied, then the set is a group. If any one of the properties is not satisfied, then the set is not a group.

What are some common examples of groups?

Some common examples of groups include the set of integers under addition, the set of real numbers excluding 0 under multiplication, and the set of all rotations of a square under composition. There are also infinite groups, such as the set of all positive integers under addition, and non-abelian groups, such as the set of all 2x2 invertible matrices under multiplication.

How is abstract algebra and group theory used in real-world applications?

Abstract algebra and group theory have many applications in fields such as physics, chemistry, computer science, and cryptography. In physics, groups are used to describe symmetries in physical systems. In chemistry, group theory is used to analyze the symmetry of molecules. In computer science, groups are used in the design of algorithms and in cryptography to ensure secure communication.

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