Proving Z=HK: Group Theory Problem Homework

  • Thread starter tyrannosaurus
  • Start date
  • Tags
    Group
In summary, the conversation discusses proving that Z=HK in group theory, where H and K are subgroups of Z. It is shown that every integer in Z can be written as a multiple of elements from H and K, indicating that Z=HK. However, it is also mentioned that the intersection of H and K is not equal to the direct product of H and K, meaning that Z is not the internal direct product of <5> and <7>. The justification for these claims may require further explanation.
  • #1
tyrannosaurus
37
0

Homework Statement


In Z, let H=<5> and K=<7>. Prove that Z=HK. Is Z the internal direct product of <5> and <7>?

Homework Equations





The Attempt at a Solution


Since 3(5)-2(7)=1, every integer n element of Z can be written as 3n(5)+(-2n)(7), and n is an element of HK( where group operation is addition and powers written as multiples). Therefore Z=HK. Since the intersection of H and K =<35>, then Z is not equal to H direct product K.
I think this is right but I am not sure on some of the justification of my claims. Can anyone help me with the justification?
 
Physics news on Phys.org
  • #2
What part are you finding a hard time justifying? You've shown Z=HK and 5^2*7^2=35*35 so it's not a direct product. What else do you need?
 

Related to Proving Z=HK: Group Theory Problem Homework

What is "Proving Z=HK" in group theory?

"Proving Z=HK" refers to a common problem in group theory where one is asked to prove that the group Z is equal to the product of two subgroups, H and K, denoted as Z=HK. This is often used as an exercise to demonstrate understanding of group theory concepts.

What is the significance of proving Z=HK?

Proving Z=HK is important because it helps to establish the relationship between different subgroups within a larger group. It also demonstrates the use of group operations, such as multiplication, to show that two subgroups can produce the entire group.

What are some strategies for solving a "Proving Z=HK" problem?

One strategy is to use the definition of Z as the set of all possible products of elements from the subgroups H and K. Then, compare this definition to the elements of Z=HK and show that they are equivalent. Another strategy is to use the properties of subgroups, such as closure, to show that Z=HK is a valid subgroup of a larger group.

Are there any common mistakes to avoid when solving a "Proving Z=HK" problem?

One common mistake is assuming that Z=HK is true without proper justification. It is important to carefully show each step of the proof and explain why it is valid. Another mistake is confusing the order of operations within the groups, which can lead to incorrect solutions.

How does solving "Proving Z=HK" relate to real-world applications?

The concept of "Proving Z=HK" can be applied to various real-world scenarios, such as cryptography, where one needs to show that a certain group operation can produce a specific result. It can also be used in the study of symmetry and geometric transformations, where groups are used to model and analyze these concepts.

Similar threads

Is N Always in the Center of G or Does It Intersect H or K Nontrivially?
    • Calculus and Beyond Homework Help
Replies
7
Views
2K
Prove if ##a\cdot c = b \cdot c## then ##a = b## using Peano postulates
    • Calculus and Beyond Homework Help
Replies
2
Views
521
Group Theory: Finite Abelian Groups - An element of order
    • Calculus and Beyond Homework Help
Replies
3
Views
2K
Order of element and order of cyclic group coincide
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Do Isomorphic Groups Have Isomorphic Centers?
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Proving that Aut(K), where K is cyclic, is abelian
    • Calculus and Beyond Homework Help
Replies
5
Views
1K
Subgroup of Index ##n## for every ##n \in \Bbb{N}##.
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Do These Functions Qualify as Group Homomorphisms?
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Show that Z_12^* and Z_8^* are isomorphic groups
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##
    • Calculus and Beyond Homework Help
Replies
1
Views
727

Hot Threads

Recent Insights

Back
Top