Which three elements are in the proper subgroup H?

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In summary, the conversation discusses a homework problem where p and q are distinct primes and H is a proper subset of integers under addition that contains exactly three elements from the set {p, p+q, pq, p^q, q^p}. The question is to determine which of the given options are the three elements in H. The correct answer is e, as proved by the fact that if c were the answer, H would contain all integers and therefore not be a proper subgroup.
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Homework Statement


Let p and q be distinct primes. Suppose that H is proper subset of the integers nd H is grou 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,q^p
c) p, p+q,pq
d) p, p^q,q^p
e)p,pq, p^q


Homework Equations



no equations for this problem

The Attempt at a Solution


The back of my textbook says the answer is e, but I thought it would be c . I don't understand why the answer is e because if a group is under addition, the additive properties of the group should be p+q . the properties for a group under multiplication would be p*q.
a:
 
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  • #2
p*q ia p added to itself q times. So obviously if p is in a subgroup of the integers with addition, then so is p*q.

It can't be c. If it were c, then it would contain co-prime integers, hence all integers, but you are told it is a *proper* subgroup.
 

1. Which three elements are in the proper subgroup H?

The three elements in the proper subgroup H depend on the specific group in question. Generally, a proper subgroup H will contain three elements that are a subset of the original group and satisfy the subgroup criteria, such as closure, associativity, and identity.

2. How is a proper subgroup H different from the original group?

A proper subgroup H is a subset of the original group that still follows the group's operation, but with a smaller number of elements. This means that not all elements of the original group are contained in the proper subgroup H.

3. Can a proper subgroup H have more than three elements?

Yes, a proper subgroup H can have more than three elements. The number of elements in a proper subgroup H can vary depending on the group and the specific subgroup criteria. However, it must have at least two elements to satisfy the subgroup criteria.

4. How is a proper subgroup H determined?

A proper subgroup H is determined by analyzing the elements and operations of the original group. It must contain a subset of the original group's elements and still satisfy the subgroup criteria, such as closure, associativity, and identity.

5. What is the significance of the proper subgroup H in mathematics?

The proper subgroup H plays a crucial role in understanding the structure of a group. It allows for the classification and analysis of groups based on their subgroups, providing valuable insights into the group's properties and relationships. Additionally, proper subgroups are used in various mathematical concepts and applications, such as group actions and symmetries.

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